CA1114534A - Arrangement for converting discrete signals in to a discrete single-sideband frequency division- multiplex-signal and vice-versa - Google Patents
Arrangement for converting discrete signals in to a discrete single-sideband frequency division- multiplex-signal and vice-versaInfo
- Publication number
- CA1114534A CA1114534A CA368,633A CA368633A CA1114534A CA 1114534 A CA1114534 A CA 1114534A CA 368633 A CA368633 A CA 368633A CA 1114534 A CA1114534 A CA 1114534A
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- Prior art keywords
- omega
- signal
- signals
- discrete
- fdm
- Prior art date
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Abstract
ABSTRACT:
TDM-FDM arrangement comprising a transfor-mation device for mutually mixing N digital-signals {xk(n)} k = 1,2,3,...N and for generating N con-version signals {sm(n)} wherein:
m = 1,2,3,...N
wherein amk represents an element of a constant value of an NxN-matrix A(N) = [amk] . Each of the conversion signals is applied to one of N sig-nal channels which each comprises a sampling rate increase element (SRI-element) and a digital filter having the transfer function Hm(.omega.) wherein:
wherein: .omega.1 arbitrary 0 ? .omega.o < ?
These signal channels are connected to an adder for adding the signals occurring in the various signal channels in order to obtain the FDM-signal. If the filter bank which is formed by the digital fil-ters in the various signal channels are described by the filtering matrix H(.omega.o) = [Hm [.omega.o + .omega.1 + (i-1) ?]]
and by the filtering matrix ]l then between the matrix A(N) and these filtering matrices the following relation exists if .omega.1 = O, and the relation A(N)T H(.omega.o) = 2IN
if .omega.1 ? O
wherein:
A(N)M represents the complex conjugate of A(N);
A(N)T represents the transposed matrix of A(N);
M represents the increase factor of the SRI-elements;
IN represents the NxN identity matrix.
TDM-FDM arrangement comprising a transfor-mation device for mutually mixing N digital-signals {xk(n)} k = 1,2,3,...N and for generating N con-version signals {sm(n)} wherein:
m = 1,2,3,...N
wherein amk represents an element of a constant value of an NxN-matrix A(N) = [amk] . Each of the conversion signals is applied to one of N sig-nal channels which each comprises a sampling rate increase element (SRI-element) and a digital filter having the transfer function Hm(.omega.) wherein:
wherein: .omega.1 arbitrary 0 ? .omega.o < ?
These signal channels are connected to an adder for adding the signals occurring in the various signal channels in order to obtain the FDM-signal. If the filter bank which is formed by the digital fil-ters in the various signal channels are described by the filtering matrix H(.omega.o) = [Hm [.omega.o + .omega.1 + (i-1) ?]]
and by the filtering matrix ]l then between the matrix A(N) and these filtering matrices the following relation exists if .omega.1 = O, and the relation A(N)T H(.omega.o) = 2IN
if .omega.1 ? O
wherein:
A(N)M represents the complex conjugate of A(N);
A(N)T represents the transposed matrix of A(N);
M represents the increase factor of the SRI-elements;
IN represents the NxN identity matrix.
Description
4S34 PHN7~
to.6.77 "Arrangement for convertlng discrete signals into a di~ision discrete single-sideband frequenc~/-multiplex-signal and vice versa."
~, .
' -A. BackFround of the inventionA(1) Field of the invention The invention relates to a discr~te single- '~ -sideband frequency-division-multiplex arrangement as ~ i -well as to a demultiplex-arrangement. ``~
In a frequency-division-multiplex arrange-meilt a plurality of baseband signals is processed suoh that they can be transmitted simultaneously in a given ~requency band. Hereinafter this frequency band will be called FDM-band. This FDM-band com-`~
prises a plurality of non-overlapping subbands. By means of some modulation process or another the fre-.
quency band o~ such a base band signal i9 shifted to :~
a subband which is characterlstic ~or the rele~ant ~!
` ' ':; , ' ~; 15 baseband signal. The signals in the ~uccessive sub- *
bands are called channel signals. The signal in the FDM-band which is constituted by all chaml~l signals will, as usual, be called FDM - signal.
: ` ~
A kncwn moduiation process is amplitude modulation, However, amplitude modulation is not economicai. ~or, with amplitude modutation, both sidebarlds o~ the modula'ed signal are transmitted.
z _ ,~
~ qP - :.
5;34 1 0 . 6 . 77 The bandwidth required for transmitting an ampl,tude-modulated signal therefor is twice as large as the -~
bandwidth which is required for the transmission o~ ~
on.y one side-band. ~-As the communication density in a tele- -communication system increased, that is to say as -more baseband signals had to be transmitted, it was desired to use the available FDM-band more ef-ficiently, As a result thereof a modulation proce~s -1G designated as single-sideband modulation was used more and more where only one sideband as the name implies is transmitted. By using single-sideband , modulation twlce as many channel slgnals can be transmitted in the given FDM-band as with ampli-tude modulatlon. It is true that with single-side-band modulation an efficient manner of transmission has been realized in terms of required bandwidth but the manner in which the slngle-sideband FDM-signal is generated must be made as simple and eco-.
nomir as is technologically possible. Particular is this true when a large number of baseband signals must be converted into a single-sideband FDM-sig~
nal. -- .
If a frequency division multi~lex arrangement is used at the transmitter side of a telecommunication system a device must be used at its receiver end for convertlng the -FDM-signal in'o the individual chan- , ~ ' PHN ~731 ~ 10.6.77 nel signals and these chamlel signals must again be -reconverted into the original baseband signals. Such a device may be called frequency-division-demultiplex "
arrangement. Also in this arrangement the oonversion of the FDU-signal into the original baseband signals ; must be as simple and economlc as is technologically possible.
A(2) D~scrlption Or the Prior art For the conversion of analog baseband sig-nals xk(t~ into an analog FDM signal y(t)~ a fre-quency-division-multiplex~arrangement might, for -.: : .
example, comprise a plurality of modulation chan- `~
; - nels each provided wi~th a single-sideband modulation oircuit. The~baseband signals xk(t) are eaoh applied 1S ~ to one of the modulation channels. The single-side~
band channél slgnals whlch are generated by these :. , modulation channels are thereafter oombined which , ;
; resulta ln the desired~SSB-FDM signal. A typlcal ~ slngle-sldeband modulation circuit for processing ,~ 20 ~ analog signals is the Weaver modulator which is described in reference 1 ~see chapter D). Analog filters are used in this known single-sideband modu-lation oircuit. Ihe rapid~develnpment of the lnte-` grated circui~t teohnology and the possibility of ~
large scale integration of discrete circuits has ~-made discrete filters~muoh more attractive than their analog counterparts. The straight forward :
: , ,.
10.6.77 .
substitution, however, Or discrete filters for analog filteræ results in a system which requires an undesir-ably high number of computational steps p~ second.
In the references 3, 4, 5 digital fre-quency-division-multiplex arrangements are describ-ed. These known arrangements are arranged for con_ ~- -verting N digital baseband signals fxk(n)~ , (k = 1,2,...N; n = ... -3, -2, -1, 0, l1, l2, ..., which are each constituted by a series of compo-nents xl(n) and which each have associated there- ;
with a sampling frequency 1/T, into a digital base- ~-band ~ingle-sideband frequency-multiplex-signal y(n)~ which is constituted by a series ofco~onents y(n) and which has associated therewith a sampling fre- -quency 1/T1 which is greater than or is equal to the frequency N/T. The signals ~ xk(n)~ can each ~ -be applied to the arrangement throngh a 4eparate lead, but also in a TDM format. For bre~ity's sake, the digital frequency-division-multiple~ arrange- ~ -ment will hereinafter be indicated by TDM-FDM ar-rangement. These arrangements can be~classified ~ ;~
into two categories:
1. The first category includes those TDM-FDM ar-rangements which each comprises a plurality of mo-dulation channels to each of which a baseband sig- ~ -nal f xk(n)} is applied. In each of these modula-tion channels a modulation processing is performed ., , I :~."
' ': ' :
10.6.77 : .i. -1~4S3~ `', ':
. 1 ,,~ ::
using-a carrier si.gnal having a carrier frequency which lS characteristic for the relevant modulation channel, as well as a single-sideband modulation ¦
processing operation whereby each modulation charmel .
generates the single-sideband modulated version of ; ~ its input signal ~xk(n)~ , the freqùency spectrum j .-~
: : o~ a typical single-sideband modulated version of -:
an input signal ~ xk(n)3 is located in a subband ~ .
. of the baseband FDM-signal, which subband is typical for the relevant baseband signal ~xk(n)} and is charac- ;`~-terized by said carrier frequency. With the l`DM-FDM- ¦ . :;.
arrangements described in references 3, 4 and 5 the :
.. : baseband signals' fxk(n)~ ~are first applied to .. ~
: an input circuit whi~h comprises means for selec- : :`
~;.15 ~ tively modulating these Dignals ~ xk(n)} for ge- . ~ .
:~ nerating discrete selectively modulated baseband ~:
: r . .
ignal8 t rk(n) ~ having associated therewith a .~.
sampling frequency 1/Tr. In particular this selec-~- tive modulatlon consists in that each real baseband :- .
; 20 signal f xk(n)~ is converted into a complex sig-nal {rk(n)~ wherein rk(n~ = Re ~rk(n)] I i Im rk(n)] and wherein the components Re [rk(n)]
: and Im [rk(n)] ocour with a sampling period `.
: ~ T/2. Thereafter these complex signals, possibly ::
after having been processed further, are modulat-ed on a complex carrier signal ha~ing a carrier frequency which is characteristic for the relevant i ::
:
~: :
`' PHN 873~
~ 534 10.6.77 ~ ~
modulation channel. For performing said selective mo- ~ -dulation and for performing said modulation on the complex carrier signal each modulation channel is constructed as a digital Weaver-modulator wherein digital modulators, as well as digital filters are used.
It is noted that in reference 5 a TDM-FDM-arrangement i9 described which is equivalent to the TDM-FDM-arrangements described in the re-ferences 3 and 4.~Particularly the TDM-FDM-arrange-ment disclosed in reference 5 comprises only one ~ ;
~:.
single-sideband modulation channel which for the ; ~ ~ varlous base-band signals ~xk(n)} is operated in tine sharing. ~ ~
~ ~ 15 This equivalency also applies to what :
; follows hereinafter.
;~ 2. The second category includes those TDM-FDM-arrangemente wherein no~ modulation processing ~ ~ , is applied utilizing a carrier signal having a car-; ~ ~ 20 rier frequency which is characteristic for the re-levant signal f xk(n)~ ~ In the TDM-FDM-arrangements of this category use is made of the properties of a discrete signal and more specifically of the fact that the frequenc~ spectrum of a discrete slgnal ha~ a periodical structure, the period being equal to the value of the sampling frequency 1/T of the baseband signal. An arrangement belonging to this ` ~
34 10.6.77 ':
second category has already been proposed in refer- ;
ence 4. It comprises N-signal channels, N being equal to the number of base-band signals. A baseband signal being applied to each of these signal channels. Each ~ , of the signal channels comprises means for increasing , the sampling rate associated with the baseband signal by a factor of N to a value N/T. By increasing the sampling rate a discrete signal ~tk(n)} is ob-tained having a periodic frequency spectrum i~
whose period is equal to, but whose fundamental ~, intervals is equal to N/T (see chapter (1.2). Each interval of the length N/T of the frequency spec-trum includes 2N subbands each having a width of 1/(2T). Each signal-channel further comprises a discrete bandpass filter having a bandwidth 1/(2T).
;.
The passbands of the bandpass filter included in the successive signal channels coincide with the successive eubbands of the first N subbands of the frequency spectrum of the discrete signal ~ tk(n)} . So the output signals {uk(n)~ of the successive bandpass filters represent the de-sired channel signals for the baseband FDM-signal.
The input circuit of this known arrange-ment also comprises means for selectively modulat-{xk(n)} . In this particular case this means that either the components of the signals f x~(n)~ having an even number k, or those having - 8 _ ::
PHN 8731 .
10,6.77 .
an odd number k are multiplied by a factor (~
The result of this multiplication is described in chapter E(1.3).
B. Summarv of the invention The invention relates to a TDM-FDM ar-rangement of the second category, which is arrang- .
: - ed for converting N discrete baseband signals xk(n)~ , (k = 1, 2~ 3~ ... N; n = 0, + 1, ~ 2 ...), having associated a sampling rate T and which each .
have a frequency spectrum ~ ( ~ ) into a discrete baseband single-sideband frequency-division-mul-tiplex-signal {y(n)} havlng associated there-; with a sampling rate T which is at least equal to .::
N and -which has a frequency spectrum Y(~)) wherein -;
~Y[~J1 ~ ~ 0 + (k-1) T] = Xk(~o) ~ k(~Jo)~ -. . It is an object of the invention to pro~
~..................................................................... . .:;~ : vide another concept of a TDM-FDM arrangement of .:, ~ : , :: : ;
: the second category and as described in chapter A(a) with which~a high degree of design freedom is ~ .
realized and which may result in a simple TDM-FDM
, . .. . . .
arrangement. .
The TDM-FDM arrangement according to the -~
invention is therefore characterized in that.for.
, ~U = o `'~
::
I. i~ comprises .
- means for receiving said baseband signals {xk(n)~ ;
:~ .
- .: .
_ 9 " .
~4534 1 o 6 77 .
- means for selectively modulating the received sig- : :
nals ~xk(n)~ for generating baseband signals ~ -, ~ ~rk(n)~ ;
- a transformation device for processing said se-~ ` 5 lectively modulated baseband signals ~rk(n)3 :~ for generating a plurality of discrete conversion ' ~ : signals f s (n)~ m = 1, 2, 3, ........ N), said ~, - . transformation device having associated therewith ., a transformation mat~ix A comprising the matrix.
' :---elements amk ~ a constant value and which trans- :~
~ : formation matrix is unequal to the Discrete Fourier ; ~ Transform (DFT) matrix, and whereby the relation . ,~:
, between the components sm(n) and the components rk(n) being given by ~: ; : N
9m(n) = ~ amk rk(n) (1) ~ - a plurality o~ Rlgnal channels to each of which a : conversion signal:is applied and which are each pro- ;
: vided with discrete filter means and sampling rate increasing means for generating discrete signals ~ um(n)~ ; the transfer function of the signal channel determined by said discrete filter means being equal to Hm(~V); .
- means for forming a digital sum signal `:
..
-- 1 0 -- ~
'`
PHN 8731 .:
10,6.77 ~4S~4 :-'`'' um(n) wherein m=1 :: :
,.
N
y(n) = ~ um(n) (2) ~
m=1 .
II. that for each signa1 channel the relation between ~ -its transfer functio~ Hm(~) and the matrix elements 5 ~ amk is given by the FDM-condition ' :.
' ~ ~ {a k Hm [~JO +(i-1) T ~ + amk m ~Ty ( ) ~ ]} ki ~i( WO) (3) ~ :
wherein~
. m is the number of the relevant signal channel;
~ : 10 ~J represents a frequency within the range :~ ; . ~ o < ~r/T; .:
~: amk denotes the oomplex oonjugate of amk; ' ~:
H~(~J) denotes the complex conjugate of H (W ~;
i = 1, 2, 3, ...N;
ki ~ for k ~ i : 15 ~ ki = 1 for k - i;
Jo) denotes an arbitrary function of The TDM-FDM arrangement according to the .
: in~ention is also characterized in that for ~
g 7r with ~ = 0, ~ 2~- ~ `
20 ~ I. it comprises: -. . ~
. PHN 8731 ~ 534 10.6.77 ~
- means for receiving said signals fxk(n)~ ;
- a cascade arrangement of selective modulation means - .:~=
and complex modulation means, whose input is coupled to said receiver means and which is arranged for ge~
~ 5 . nerating complex signals ~rk(nj3 with which com- : :
: plex modulation means a complex carrier aignal hav- -:--ing a frequency 21r is associated;
. . - a transformation device for processing said sig- :
~, nals ~rk(n)~ and for generating a plurality of.
~ ~10 ~ discrete conversion signals fs (n)~
: (m = 1, 2, 3, ... N) said transformation devioe hav-: ing associated therewith a transformation matrix A
; comprising bhe elements amk of a constant value ~ - and whioh transformation matrix is unequal to the ; ~ 15 ~Discrete Fourier Transform (DFT)-matrix and whereby the relation between the oomponents sm(n) and the components rk(n) is given by . ~ .
~ sm(n) = ~ amk rk( ) : :
. -a Rlurality of signal ohannels, to each of which a conversion signal is applied and which are each : provided with discrete filter means and sampling .
~ rate - increasing means for generating discrete ~ ~
,~
. signals tu (n) , , the transfer function of the `~
signal channel determined by said discrete filter means being equal to Hm(~));
, .
-:
.. . .. . . .. ....... . . . . . . . . . .. . .. . . . . .
PHN 8 7 3 1 ' - ' ~1~4~34 10.6.77 ".,~ ,'.
- means for forming a discrete sum signal ,~ ~ ~ u (n)~ whereln y(n) = ~ u (n);
II. that for eaoh signal channel the relation between its transfer function H (t~) and the matrix elements 5 ~ ~ amX i8 given by the FDM-condition m [ T ~ f ~0 ~ W 1 + (i-1) T-~ = ~ ~ -a~kH ~0 + ~J1 + (~i-1) T ~ = ~ki p i( o) whereln: :.
m denotes the number of the~:relevant signal:channel; ~ .~
'.~ 10 ~ ~J representD a frequency within the range ~ - ~j:
0 ~ <. ; ,: , Hm ( W ) denotes the complex con~ugate of Hm(~ ); :
i = 1, 2, 3~ .... N; ~~
ki = for k ~ i : :;~
" .
~:~ ki 1 for k = i;
~i( ~0) denotes an arbitrary function of ~J0.
~ The invention also relates to an FDM-TDM- - -. arrangement for ~onverting a d-screte baseband ~
single-sideband frequency divislon-multiplex sig- ~.
nal f y(n)~ , (n = 0, ~ 2, ~ 3...) having as-sociated therewith a sampling rate 1/Ty which is :
at leasb equal to N/T and which has a frequency :
'; . ,, .:; ~ ''' .' ~'~' .
10.6.77 :-1$~4534 ~ :~
,~:
spectrum Y(~J) into a number of N discrete baseband ~.
signals lxk(n)} , (k =-1, 2, 3,...) each having as-sociated therewith a sampling rate 1/T and which each .-.
have a frequency spectrum ~ (~J) wherein [ 0 1 ~ (k~ k( W0)- -~
It is an other object of the invention to provide a FDM-TDM-arrangement having a high degree ! .
: of design freedom, so that a simple.FDM-TDM arrange- .
ment can be obtained. ~ ¦
The FDM-TDM-device according to the inven-tion is therefore characterized in that. for ~J = O .
~ 1 I. it comprises: .
- means for receiving said discrete frequency divi.sion-multiplex-signal ~ y(n)~ ; ~ ;
~: 15 ~ - a plurality of signal channels to each of which .
, , , .
: said discrete frequency-division-multiplex signal ~.
{ y(n)~ is applied and which are each provided with dlscrete filter means and sampling rate re- ~
~ duction means for generating discrete signals :
.~ 20 f~m(n)~ ; the .transfer function of the signal -~
channel determined by said filter means being .:-equal to Em(W );
- a transformation device to which said discrete : signals ~ sm(n)~ are applled and which is arranged for processing these signals for generating a plu-rality of discrete signals frk(n)~ ; said trans-:~ -' ' , . :,.
: - 14 -~.': , ~" :
PHN 8731 :~
~ S34 10.6.77 ' formation device having associated therewith a trans-formation matrix B comprising the matrix elements bkm of a constant value and which transformation matrix . ~
i8 unequal to the Inverse Discrete Fourier Trans- :
form (IDFT) matrix and whereby the relation between the components sm(n) and the components rk(n) is '-. given by: - .
, i,.: . ...
.: ,:.
N .- :
rk(n) = ~ bkm 9m( ) ~ an output circuit bo which the signals {rk(n)~
:~ 10 ~ are applied and which is provided with means for ~.
selcctively modulating the signals ~rk(n)} for generabing said discrete~baseband signals f xk ( n ) ,~
II. that for each slgnal channel the relation be-tween its transfer function Em(~) and the matrix elements bkm is given by the TDM-condition N ~ {b E ~ (i 1) ~ ] ~ :
[ T f o ~ T } ]3 s ki ~i ( ~o, : ~
wherein:
m represents the number of the relevant signal channel;
~J represents a frequency within the range :
.
.
10.6.77 .~i 4S34 `
~ ~ -~ o ~ T
bkm denotes the complex conjugate of bkm;
Em (~l) denotes the complex conjugate of Em( W );
i = 1, 2, 3, ...N
~ki 0 for k ~ i ~ki = 1 for k = 1 ~i ( WO) denotes an arbitrary function of ~0. ;
' The FDM-TDM arrangement according to the in- ~ :~
vention i8 also characterized in that for W 1 ~ ~ T ~
~:~ 10 with ~ = 0, +1, +2,.......................................... .
I. it comprises:
means for receiving said frequency-division-mul-: tiplex-signal {y(n)~
- a plurality of signal channels to each of which ~15 said discrete frequency-division-multiplex-signal y(n)~ is appliëd and each comprising discrete filter means and sampling rate reduction means ~ :
for generating discrete signals fsm(n)} ; the transfer function of the signal channel determined by said filter means being equal to E (~J);
- a transformation device to which said discrete signals {s (n)} are applied and which is arrang-ed for processing these signals for generating a plurality of discrete signals frk(n)~ , said ~:
transformation device . -~
~ 34 1 o. 6.77 having associated therewith a transformation matrix B comprising the matrix elements bkm of a constant value, said transformation matrix being uneaual .~
to the Inverse Discrete Fourier Transform (IDFT) ~;:
matrix, whereby the relation between the components ~ :
sm(n) and the oomponents rk(n) iY given by `:
:~ ` rk(n) ~ ~ bkm =m( ) ...
- an output circult to which the signals ~rk(n)~
~1 are applied and which comprises a cascade arrange~
ment oP selective modulation means and complex mo-dulatlon means with wh ch a complex carrier signal having the frequency ~ lS associated, for generat~
:~ ing said dlscrete baseband signals ~ xk(n)} ;~' ' II. that for each.signal channel the relation be- :
tween the trans~er function Em( ~) and the matrix elements bkm is given by the TDM-condition m ~ T ~ f~ ~ ~J0 ~ 1) T-~ ] ~
2N ~ bk~Em ~1 + W 0 ~ l) T~ ]
:: :':
. ~ kl y i( W0). ~ ~ :
wherein:
m Fepresents the number of the relevant signal:
-- - 17 _ ~ ' ~
10.6.77 3~
channel;
W represents a frequency within the range O ~ ~1o~ T ; ;
Em ( W ) denotes the complex conJugate of Em(W )-i = 1, 2, 3, ...... N ;
6 ki = for k ~
~ki = 1 for k = i .: -, .
J0) denotes an arbitrary function of hJo.
In order to avoid interchannel interfer~
ence very complex dlscrete filters must be used in the arrangements described .in chapter A(2).
By means of, for example, the instruction .~
; given in expression (3) and by using the further :;-measures according to the invention it is possible ~: ~15 to realize an optimum distribution of the circuit complexity over the transformation device and the discrete filter means. More in particular, many applications, such as the conversion of a number of signalling signals into an FDM-format allow a particularly simple transformation device together with simple discre.te filter means, If the matrix ; elements amk are each given by ~ mk + i~
then the values of the constants ~ k and ~ k -may, for example, be given by the set of numbers (0, +1, -1)- .
It should be noted that in references `
Pl-IN 87~1 -10.6.77 '~
34 ::
: . , . .,:
6 to 10 inclusive TDM-FDM arrangements as well as ,~
FDM-TDM arrangements are described which must be - -~
considered as special embodiments of the generic arrangement according to the invention. In these special embodiment~ a transformation device is used ;
in the form of a DFT (Discrete Fourier Transformer), so that the matrix elements a k are equal to exp [ - 2 ~ j(m-1)(k-1)/N] wherein N again re-presents the number of baseband signals f xk(n)3 .
The instruction which, for example, is defined in expression (3) is not lndicated in these references ~' ;~ ~ and it is also not possible to derive them therefrom. -These references also do neither show nor suggest ~ that other values of the matrix elements may be ;~
;~ 15 used. Modifications of the TDM-FDM arrangements ~ ;
; as well as modifications of the FDM-TDM-arrangement, described in next chapters are not indicated in these references and cannot be deduced therefrom.
The TDM-FDM arrangement according to the invention the TDM-~DM arrangement described in the re~erences 6 to 1~ inclusive have in common that the baseband signals ~ xk(n)~ are not processed -~
separately as in the second class TDM-FDM arrange-ment described in reference 4, but that they are ~ -mutually mixed in a transformer.
~It should be noted that a discrete signal -~
is defined as a signal which ~s exclusively defined 10.6.77 534 ~ ~
at discrete time instants (see reference 2). These signals can be classified in two categories, namely: `
1. Digital signals. These signals are discrete time signals that can take on discrete amplitude values.
Such signals are available in the form of a series Or numbers whioh are each represented by a given number of bits. ;
to.6.77 "Arrangement for convertlng discrete signals into a di~ision discrete single-sideband frequenc~/-multiplex-signal and vice versa."
~, .
' -A. BackFround of the inventionA(1) Field of the invention The invention relates to a discr~te single- '~ -sideband frequency-division-multiplex arrangement as ~ i -well as to a demultiplex-arrangement. ``~
In a frequency-division-multiplex arrange-meilt a plurality of baseband signals is processed suoh that they can be transmitted simultaneously in a given ~requency band. Hereinafter this frequency band will be called FDM-band. This FDM-band com-`~
prises a plurality of non-overlapping subbands. By means of some modulation process or another the fre-.
quency band o~ such a base band signal i9 shifted to :~
a subband which is characterlstic ~or the rele~ant ~!
` ' ':; , ' ~; 15 baseband signal. The signals in the ~uccessive sub- *
bands are called channel signals. The signal in the FDM-band which is constituted by all chaml~l signals will, as usual, be called FDM - signal.
: ` ~
A kncwn moduiation process is amplitude modulation, However, amplitude modulation is not economicai. ~or, with amplitude modutation, both sidebarlds o~ the modula'ed signal are transmitted.
z _ ,~
~ qP - :.
5;34 1 0 . 6 . 77 The bandwidth required for transmitting an ampl,tude-modulated signal therefor is twice as large as the -~
bandwidth which is required for the transmission o~ ~
on.y one side-band. ~-As the communication density in a tele- -communication system increased, that is to say as -more baseband signals had to be transmitted, it was desired to use the available FDM-band more ef-ficiently, As a result thereof a modulation proce~s -1G designated as single-sideband modulation was used more and more where only one sideband as the name implies is transmitted. By using single-sideband , modulation twlce as many channel slgnals can be transmitted in the given FDM-band as with ampli-tude modulatlon. It is true that with single-side-band modulation an efficient manner of transmission has been realized in terms of required bandwidth but the manner in which the slngle-sideband FDM-signal is generated must be made as simple and eco-.
nomir as is technologically possible. Particular is this true when a large number of baseband signals must be converted into a single-sideband FDM-sig~
nal. -- .
If a frequency division multi~lex arrangement is used at the transmitter side of a telecommunication system a device must be used at its receiver end for convertlng the -FDM-signal in'o the individual chan- , ~ ' PHN ~731 ~ 10.6.77 nel signals and these chamlel signals must again be -reconverted into the original baseband signals. Such a device may be called frequency-division-demultiplex "
arrangement. Also in this arrangement the oonversion of the FDU-signal into the original baseband signals ; must be as simple and economlc as is technologically possible.
A(2) D~scrlption Or the Prior art For the conversion of analog baseband sig-nals xk(t~ into an analog FDM signal y(t)~ a fre-quency-division-multiplex~arrangement might, for -.: : .
example, comprise a plurality of modulation chan- `~
; - nels each provided wi~th a single-sideband modulation oircuit. The~baseband signals xk(t) are eaoh applied 1S ~ to one of the modulation channels. The single-side~
band channél slgnals whlch are generated by these :. , modulation channels are thereafter oombined which , ;
; resulta ln the desired~SSB-FDM signal. A typlcal ~ slngle-sldeband modulation circuit for processing ,~ 20 ~ analog signals is the Weaver modulator which is described in reference 1 ~see chapter D). Analog filters are used in this known single-sideband modu-lation oircuit. Ihe rapid~develnpment of the lnte-` grated circui~t teohnology and the possibility of ~
large scale integration of discrete circuits has ~-made discrete filters~muoh more attractive than their analog counterparts. The straight forward :
: , ,.
10.6.77 .
substitution, however, Or discrete filters for analog filteræ results in a system which requires an undesir-ably high number of computational steps p~ second.
In the references 3, 4, 5 digital fre-quency-division-multiplex arrangements are describ-ed. These known arrangements are arranged for con_ ~- -verting N digital baseband signals fxk(n)~ , (k = 1,2,...N; n = ... -3, -2, -1, 0, l1, l2, ..., which are each constituted by a series of compo-nents xl(n) and which each have associated there- ;
with a sampling frequency 1/T, into a digital base- ~-band ~ingle-sideband frequency-multiplex-signal y(n)~ which is constituted by a series ofco~onents y(n) and which has associated therewith a sampling fre- -quency 1/T1 which is greater than or is equal to the frequency N/T. The signals ~ xk(n)~ can each ~ -be applied to the arrangement throngh a 4eparate lead, but also in a TDM format. For bre~ity's sake, the digital frequency-division-multiple~ arrange- ~ -ment will hereinafter be indicated by TDM-FDM ar-rangement. These arrangements can be~classified ~ ;~
into two categories:
1. The first category includes those TDM-FDM ar-rangements which each comprises a plurality of mo-dulation channels to each of which a baseband sig- ~ -nal f xk(n)} is applied. In each of these modula-tion channels a modulation processing is performed ., , I :~."
' ': ' :
10.6.77 : .i. -1~4S3~ `', ':
. 1 ,,~ ::
using-a carrier si.gnal having a carrier frequency which lS characteristic for the relevant modulation channel, as well as a single-sideband modulation ¦
processing operation whereby each modulation charmel .
generates the single-sideband modulated version of ; ~ its input signal ~xk(n)~ , the freqùency spectrum j .-~
: : o~ a typical single-sideband modulated version of -:
an input signal ~ xk(n)3 is located in a subband ~ .
. of the baseband FDM-signal, which subband is typical for the relevant baseband signal ~xk(n)} and is charac- ;`~-terized by said carrier frequency. With the l`DM-FDM- ¦ . :;.
arrangements described in references 3, 4 and 5 the :
.. : baseband signals' fxk(n)~ ~are first applied to .. ~
: an input circuit whi~h comprises means for selec- : :`
~;.15 ~ tively modulating these Dignals ~ xk(n)} for ge- . ~ .
:~ nerating discrete selectively modulated baseband ~:
: r . .
ignal8 t rk(n) ~ having associated therewith a .~.
sampling frequency 1/Tr. In particular this selec-~- tive modulatlon consists in that each real baseband :- .
; 20 signal f xk(n)~ is converted into a complex sig-nal {rk(n)~ wherein rk(n~ = Re ~rk(n)] I i Im rk(n)] and wherein the components Re [rk(n)]
: and Im [rk(n)] ocour with a sampling period `.
: ~ T/2. Thereafter these complex signals, possibly ::
after having been processed further, are modulat-ed on a complex carrier signal ha~ing a carrier frequency which is characteristic for the relevant i ::
:
~: :
`' PHN 873~
~ 534 10.6.77 ~ ~
modulation channel. For performing said selective mo- ~ -dulation and for performing said modulation on the complex carrier signal each modulation channel is constructed as a digital Weaver-modulator wherein digital modulators, as well as digital filters are used.
It is noted that in reference 5 a TDM-FDM-arrangement i9 described which is equivalent to the TDM-FDM-arrangements described in the re-ferences 3 and 4.~Particularly the TDM-FDM-arrange-ment disclosed in reference 5 comprises only one ~ ;
~:.
single-sideband modulation channel which for the ; ~ ~ varlous base-band signals ~xk(n)} is operated in tine sharing. ~ ~
~ ~ 15 This equivalency also applies to what :
; follows hereinafter.
;~ 2. The second category includes those TDM-FDM-arrangemente wherein no~ modulation processing ~ ~ , is applied utilizing a carrier signal having a car-; ~ ~ 20 rier frequency which is characteristic for the re-levant signal f xk(n)~ ~ In the TDM-FDM-arrangements of this category use is made of the properties of a discrete signal and more specifically of the fact that the frequenc~ spectrum of a discrete slgnal ha~ a periodical structure, the period being equal to the value of the sampling frequency 1/T of the baseband signal. An arrangement belonging to this ` ~
34 10.6.77 ':
second category has already been proposed in refer- ;
ence 4. It comprises N-signal channels, N being equal to the number of base-band signals. A baseband signal being applied to each of these signal channels. Each ~ , of the signal channels comprises means for increasing , the sampling rate associated with the baseband signal by a factor of N to a value N/T. By increasing the sampling rate a discrete signal ~tk(n)} is ob-tained having a periodic frequency spectrum i~
whose period is equal to, but whose fundamental ~, intervals is equal to N/T (see chapter (1.2). Each interval of the length N/T of the frequency spec-trum includes 2N subbands each having a width of 1/(2T). Each signal-channel further comprises a discrete bandpass filter having a bandwidth 1/(2T).
;.
The passbands of the bandpass filter included in the successive signal channels coincide with the successive eubbands of the first N subbands of the frequency spectrum of the discrete signal ~ tk(n)} . So the output signals {uk(n)~ of the successive bandpass filters represent the de-sired channel signals for the baseband FDM-signal.
The input circuit of this known arrange-ment also comprises means for selectively modulat-{xk(n)} . In this particular case this means that either the components of the signals f x~(n)~ having an even number k, or those having - 8 _ ::
PHN 8731 .
10,6.77 .
an odd number k are multiplied by a factor (~
The result of this multiplication is described in chapter E(1.3).
B. Summarv of the invention The invention relates to a TDM-FDM ar-rangement of the second category, which is arrang- .
: - ed for converting N discrete baseband signals xk(n)~ , (k = 1, 2~ 3~ ... N; n = 0, + 1, ~ 2 ...), having associated a sampling rate T and which each .
have a frequency spectrum ~ ( ~ ) into a discrete baseband single-sideband frequency-division-mul-tiplex-signal {y(n)} havlng associated there-; with a sampling rate T which is at least equal to .::
N and -which has a frequency spectrum Y(~)) wherein -;
~Y[~J1 ~ ~ 0 + (k-1) T] = Xk(~o) ~ k(~Jo)~ -. . It is an object of the invention to pro~
~..................................................................... . .:;~ : vide another concept of a TDM-FDM arrangement of .:, ~ : , :: : ;
: the second category and as described in chapter A(a) with which~a high degree of design freedom is ~ .
realized and which may result in a simple TDM-FDM
, . .. . . .
arrangement. .
The TDM-FDM arrangement according to the -~
invention is therefore characterized in that.for.
, ~U = o `'~
::
I. i~ comprises .
- means for receiving said baseband signals {xk(n)~ ;
:~ .
- .: .
_ 9 " .
~4534 1 o 6 77 .
- means for selectively modulating the received sig- : :
nals ~xk(n)~ for generating baseband signals ~ -, ~ ~rk(n)~ ;
- a transformation device for processing said se-~ ` 5 lectively modulated baseband signals ~rk(n)3 :~ for generating a plurality of discrete conversion ' ~ : signals f s (n)~ m = 1, 2, 3, ........ N), said ~, - . transformation device having associated therewith ., a transformation mat~ix A comprising the matrix.
' :---elements amk ~ a constant value and which trans- :~
~ : formation matrix is unequal to the Discrete Fourier ; ~ Transform (DFT) matrix, and whereby the relation . ,~:
, between the components sm(n) and the components rk(n) being given by ~: ; : N
9m(n) = ~ amk rk(n) (1) ~ - a plurality o~ Rlgnal channels to each of which a : conversion signal:is applied and which are each pro- ;
: vided with discrete filter means and sampling rate increasing means for generating discrete signals ~ um(n)~ ; the transfer function of the signal channel determined by said discrete filter means being equal to Hm(~V); .
- means for forming a digital sum signal `:
..
-- 1 0 -- ~
'`
PHN 8731 .:
10,6.77 ~4S~4 :-'`'' um(n) wherein m=1 :: :
,.
N
y(n) = ~ um(n) (2) ~
m=1 .
II. that for each signa1 channel the relation between ~ -its transfer functio~ Hm(~) and the matrix elements 5 ~ amk is given by the FDM-condition ' :.
' ~ ~ {a k Hm [~JO +(i-1) T ~ + amk m ~Ty ( ) ~ ]} ki ~i( WO) (3) ~ :
wherein~
. m is the number of the relevant signal channel;
~ : 10 ~J represents a frequency within the range :~ ; . ~ o < ~r/T; .:
~: amk denotes the oomplex oonjugate of amk; ' ~:
H~(~J) denotes the complex conjugate of H (W ~;
i = 1, 2, 3, ...N;
ki ~ for k ~ i : 15 ~ ki = 1 for k - i;
Jo) denotes an arbitrary function of The TDM-FDM arrangement according to the .
: in~ention is also characterized in that for ~
g 7r with ~ = 0, ~ 2~- ~ `
20 ~ I. it comprises: -. . ~
. PHN 8731 ~ 534 10.6.77 ~
- means for receiving said signals fxk(n)~ ;
- a cascade arrangement of selective modulation means - .:~=
and complex modulation means, whose input is coupled to said receiver means and which is arranged for ge~
~ 5 . nerating complex signals ~rk(nj3 with which com- : :
: plex modulation means a complex carrier aignal hav- -:--ing a frequency 21r is associated;
. . - a transformation device for processing said sig- :
~, nals ~rk(n)~ and for generating a plurality of.
~ ~10 ~ discrete conversion signals fs (n)~
: (m = 1, 2, 3, ... N) said transformation devioe hav-: ing associated therewith a transformation matrix A
; comprising bhe elements amk of a constant value ~ - and whioh transformation matrix is unequal to the ; ~ 15 ~Discrete Fourier Transform (DFT)-matrix and whereby the relation between the oomponents sm(n) and the components rk(n) is given by . ~ .
~ sm(n) = ~ amk rk( ) : :
. -a Rlurality of signal ohannels, to each of which a conversion signal is applied and which are each : provided with discrete filter means and sampling .
~ rate - increasing means for generating discrete ~ ~
,~
. signals tu (n) , , the transfer function of the `~
signal channel determined by said discrete filter means being equal to Hm(~));
, .
-:
.. . .. . . .. ....... . . . . . . . . . .. . .. . . . . .
PHN 8 7 3 1 ' - ' ~1~4~34 10.6.77 ".,~ ,'.
- means for forming a discrete sum signal ,~ ~ ~ u (n)~ whereln y(n) = ~ u (n);
II. that for eaoh signal channel the relation between its transfer function H (t~) and the matrix elements 5 ~ ~ amX i8 given by the FDM-condition m [ T ~ f ~0 ~ W 1 + (i-1) T-~ = ~ ~ -a~kH ~0 + ~J1 + (~i-1) T ~ = ~ki p i( o) whereln: :.
m denotes the number of the~:relevant signal:channel; ~ .~
'.~ 10 ~ ~J representD a frequency within the range ~ - ~j:
0 ~ <. ; ,: , Hm ( W ) denotes the complex con~ugate of Hm(~ ); :
i = 1, 2, 3~ .... N; ~~
ki = for k ~ i : :;~
" .
~:~ ki 1 for k = i;
~i( ~0) denotes an arbitrary function of ~J0.
~ The invention also relates to an FDM-TDM- - -. arrangement for ~onverting a d-screte baseband ~
single-sideband frequency divislon-multiplex sig- ~.
nal f y(n)~ , (n = 0, ~ 2, ~ 3...) having as-sociated therewith a sampling rate 1/Ty which is :
at leasb equal to N/T and which has a frequency :
'; . ,, .:; ~ ''' .' ~'~' .
10.6.77 :-1$~4534 ~ :~
,~:
spectrum Y(~J) into a number of N discrete baseband ~.
signals lxk(n)} , (k =-1, 2, 3,...) each having as-sociated therewith a sampling rate 1/T and which each .-.
have a frequency spectrum ~ (~J) wherein [ 0 1 ~ (k~ k( W0)- -~
It is an other object of the invention to provide a FDM-TDM-arrangement having a high degree ! .
: of design freedom, so that a simple.FDM-TDM arrange- .
ment can be obtained. ~ ¦
The FDM-TDM-device according to the inven-tion is therefore characterized in that. for ~J = O .
~ 1 I. it comprises: .
- means for receiving said discrete frequency divi.sion-multiplex-signal ~ y(n)~ ; ~ ;
~: 15 ~ - a plurality of signal channels to each of which .
, , , .
: said discrete frequency-division-multiplex signal ~.
{ y(n)~ is applied and which are each provided with dlscrete filter means and sampling rate re- ~
~ duction means for generating discrete signals :
.~ 20 f~m(n)~ ; the .transfer function of the signal -~
channel determined by said filter means being .:-equal to Em(W );
- a transformation device to which said discrete : signals ~ sm(n)~ are applled and which is arranged for processing these signals for generating a plu-rality of discrete signals frk(n)~ ; said trans-:~ -' ' , . :,.
: - 14 -~.': , ~" :
PHN 8731 :~
~ S34 10.6.77 ' formation device having associated therewith a trans-formation matrix B comprising the matrix elements bkm of a constant value and which transformation matrix . ~
i8 unequal to the Inverse Discrete Fourier Trans- :
form (IDFT) matrix and whereby the relation between the components sm(n) and the components rk(n) is '-. given by: - .
, i,.: . ...
.: ,:.
N .- :
rk(n) = ~ bkm 9m( ) ~ an output circuit bo which the signals {rk(n)~
:~ 10 ~ are applied and which is provided with means for ~.
selcctively modulating the signals ~rk(n)} for generabing said discrete~baseband signals f xk ( n ) ,~
II. that for each slgnal channel the relation be-tween its transfer function Em(~) and the matrix elements bkm is given by the TDM-condition N ~ {b E ~ (i 1) ~ ] ~ :
[ T f o ~ T } ]3 s ki ~i ( ~o, : ~
wherein:
m represents the number of the relevant signal channel;
~J represents a frequency within the range :
.
.
10.6.77 .~i 4S34 `
~ ~ -~ o ~ T
bkm denotes the complex conjugate of bkm;
Em (~l) denotes the complex conjugate of Em( W );
i = 1, 2, 3, ...N
~ki 0 for k ~ i ~ki = 1 for k = 1 ~i ( WO) denotes an arbitrary function of ~0. ;
' The FDM-TDM arrangement according to the in- ~ :~
vention i8 also characterized in that for W 1 ~ ~ T ~
~:~ 10 with ~ = 0, +1, +2,.......................................... .
I. it comprises:
means for receiving said frequency-division-mul-: tiplex-signal {y(n)~
- a plurality of signal channels to each of which ~15 said discrete frequency-division-multiplex-signal y(n)~ is appliëd and each comprising discrete filter means and sampling rate reduction means ~ :
for generating discrete signals fsm(n)} ; the transfer function of the signal channel determined by said filter means being equal to E (~J);
- a transformation device to which said discrete signals {s (n)} are applied and which is arrang-ed for processing these signals for generating a plurality of discrete signals frk(n)~ , said ~:
transformation device . -~
~ 34 1 o. 6.77 having associated therewith a transformation matrix B comprising the matrix elements bkm of a constant value, said transformation matrix being uneaual .~
to the Inverse Discrete Fourier Transform (IDFT) ~;:
matrix, whereby the relation between the components ~ :
sm(n) and the oomponents rk(n) iY given by `:
:~ ` rk(n) ~ ~ bkm =m( ) ...
- an output circult to which the signals ~rk(n)~
~1 are applied and which comprises a cascade arrange~
ment oP selective modulation means and complex mo-dulatlon means with wh ch a complex carrier signal having the frequency ~ lS associated, for generat~
:~ ing said dlscrete baseband signals ~ xk(n)} ;~' ' II. that for each.signal channel the relation be- :
tween the trans~er function Em( ~) and the matrix elements bkm is given by the TDM-condition m ~ T ~ f~ ~ ~J0 ~ 1) T-~ ] ~
2N ~ bk~Em ~1 + W 0 ~ l) T~ ]
:: :':
. ~ kl y i( W0). ~ ~ :
wherein:
m Fepresents the number of the relevant signal:
-- - 17 _ ~ ' ~
10.6.77 3~
channel;
W represents a frequency within the range O ~ ~1o~ T ; ;
Em ( W ) denotes the complex conJugate of Em(W )-i = 1, 2, 3, ...... N ;
6 ki = for k ~
~ki = 1 for k = i .: -, .
J0) denotes an arbitrary function of hJo.
In order to avoid interchannel interfer~
ence very complex dlscrete filters must be used in the arrangements described .in chapter A(2).
By means of, for example, the instruction .~
; given in expression (3) and by using the further :;-measures according to the invention it is possible ~: ~15 to realize an optimum distribution of the circuit complexity over the transformation device and the discrete filter means. More in particular, many applications, such as the conversion of a number of signalling signals into an FDM-format allow a particularly simple transformation device together with simple discre.te filter means, If the matrix ; elements amk are each given by ~ mk + i~
then the values of the constants ~ k and ~ k -may, for example, be given by the set of numbers (0, +1, -1)- .
It should be noted that in references `
Pl-IN 87~1 -10.6.77 '~
34 ::
: . , . .,:
6 to 10 inclusive TDM-FDM arrangements as well as ,~
FDM-TDM arrangements are described which must be - -~
considered as special embodiments of the generic arrangement according to the invention. In these special embodiment~ a transformation device is used ;
in the form of a DFT (Discrete Fourier Transformer), so that the matrix elements a k are equal to exp [ - 2 ~ j(m-1)(k-1)/N] wherein N again re-presents the number of baseband signals f xk(n)3 .
The instruction which, for example, is defined in expression (3) is not lndicated in these references ~' ;~ ~ and it is also not possible to derive them therefrom. -These references also do neither show nor suggest ~ that other values of the matrix elements may be ;~
;~ 15 used. Modifications of the TDM-FDM arrangements ~ ;
; as well as modifications of the FDM-TDM-arrangement, described in next chapters are not indicated in these references and cannot be deduced therefrom.
The TDM-FDM arrangement according to the invention the TDM-~DM arrangement described in the re~erences 6 to 1~ inclusive have in common that the baseband signals ~ xk(n)~ are not processed -~
separately as in the second class TDM-FDM arrange-ment described in reference 4, but that they are ~ -mutually mixed in a transformer.
~It should be noted that a discrete signal -~
is defined as a signal which ~s exclusively defined 10.6.77 534 ~ ~
at discrete time instants (see reference 2). These signals can be classified in two categories, namely: `
1. Digital signals. These signals are discrete time signals that can take on discrete amplitude values.
Such signals are available in the form of a series Or numbers whioh are each represented by a given number of bits. ;
2, "Sampled data" signals. These signals are discrete tlme ~ignals that can take on a continuum of ampli-; tude values. For storing such signals "charge -~
; coupled devices" (CCD's) are, for example, used. -~
Arrangements which are suitable for pro-~- cessing digital signals are called digital arrange-ments whilst arrangements which are suitable for processing sampled data signals are indicated cor-respondingly as sampled data arrangements.
What follow~ hereinafter applies to both a digital and to a sampled data arrangement and the invention will be further explained with reference to a digital TDM-FDM arrangement and a digital FDM-TDM arrangement. -;
C. Short description of the Fi~ures.
Figure 1 shows the grafical representa- - -tion of a digital signal Figure 2 shows schematically the fre-quency spectrum of a digital si~nal; -:
-~ ;
10.6.7~
i34 . ~
' ~:
`. ;.
Figure 3 shows the symbol for a device for increasing the sampling rate (SRI-element);
Figures 4 and 5 show diagrams for explain-ing the operation of the SRI-element;
Figure 6 shows the symbol of a device for reducing the sampling rate ( ~ -element) and the figures 7, 8, 9 and 10 show some diagrams for ex-plaining the operation of the SRR-element; :.
Figure 11 shows the symbol of a sideband~
interchanging modulator; ~.
Figure 11a shows an embodiment of a side~
band interohanging modulator; .
The Figures 12, 13, 14 15 show some dia- ~
.
grams for explaining the operation of the sideband ;~.
interchanging modulator;
Figure 16 shows the symbol for a oomplex modulator and .
Figure 17 shows an embodiment of such a .. ~
.. .
modulator; whilst the Figures 18a, 18b and 18c show some frequency spectra of the signals pro-duced by the complex modulator;
Figure 19 shows the frequency spectrum ~of a real input signal of the TDM-FDM arrangement;
Figures 20a and 20b show two possible fre-quency spectra of the real SSB-FDM signal;
Figures 21 shows the bacic diagram of the TDM-FDM arrangement according to the invention; ` ~ :
. ~
,. '~
Figure 22, which is on the same sheet as ` -Fig. 20a and 20b, shows the frequency spectra of some ;-input signals of the transformer in the TDM-FDM
arrangement according to the invention;
Figure 23 shows an embodiment of a two-point transformer for real input signals and complex multiplication factors and Figure 25, on the same sheet, shows the symbol of such a transformer;
Figure 24 shows an embodiment of a two-point transformer for complex input signals and complex multiplication factors and Figure 26, which is on the ;~
same sheet as Fig. 23 and Fig. 25, shows the symbol of such a transformer; -Figure 27 shows an eight-point fast trans-former for use in the TDM-FDM arrangement shown in Figure 21;
Figure 28 shows the general set-up of a signal channel of the device according to Figure 21 and the Figures 28a, 28b and 28c show modified versions of the signal channel of Figure 28;
Figures 29 and 30 show modified versions of -the TDM-FDM arrangement shown in Figure 21;
Figure 31, which is on the same sheet as Fig. 29, shows a single circuit configuration which is used for the set-up of the signal channels in the arrangement according to Figure 30;
The Figures 32, 33, 35 and 38 and 40 show -some modifications of the circuit configuration shown in Figure 31;
.
: ~:
PI~N 8731 10.6.77 The Figures 34, 36, 37, 41, 42 and 43 show -.
some transfer functi.ons of filter means which are used in the circuit configurations of the Figures :
32, 33, 35, 38 and 40;
S ~igure 39 shows a detailed embodiment of .~
a TDM-FDM arrangement wherein the transformer is -.- ;
based on a Hadamard matrix and wherein real sig-nals are applied to the transformer; ~ .
. Figure 44 shows an embodiment of a TDM-~DM; .
arrangement wherein the transformer is based on a complex matrix and to which complex signals are .
applied;
The Figures 45 and 46 show some transfer functions of the digital filters used in the de-vice of Figure 44;
Figure 47 shows an embodiment of an FDM-TDM arrangement.
D. References 1. A third method of generation and detection of single-sideband signals; D.K. Weaver~ Proceedings of the IRE, December 1956, pages 1703 - 1705.
2. Terminology in digital signal processing;
L.R. Rabiner c.s.; IEEE transactions on audio : .
and electroacoustics, vol. AU-20, No. 5, December 1972, pages 322 - 337. .
. On digital single-sideband modulators; S~.
Darlington; IEEE transaotions on circuit theory; .
- 23 - ~
.
~ 534 ~HN ~731 vol. CT-17, No. 3, August 1970, pages 409 - 414.
4. Design of digital filters for an all digital fre-quency-division multiplex time-division multiplex translator; S.L. Freeny c.s.; IEEE transactions on circuit theory, vol. CT-18, No. 6, November 1971, pages 702 - 710.
5. SSB/FDM utilizing TDM digital filters; C.F. Kurth~
IEEE transactions on communication technology, vol.
COM-l9, No. 1, February 1971, pages 63 - 71.
6. TDM-FDM transmultiplexer; digital polyphase and FFT; M.G. Bellanger, J.L. Daguet; IEEE transactions on communications, vol. COM-22, No. 9, September 1974, pages 1199-1205. ~ -7. Single-sideband system for digital processing ~ `
15 of a plurality of channel signals; Canadian Patent -987,742 which issued to Telecommunications Radio-electriques et Telephoniques T.R.T. on April 20, 1976. -8. Circuit arrangement for digitally processing a given number of channel signals; Canadian Patent 1,025,135 which issued to Telecommunications Radio-electriques et Telephoniques T.R.T. on Jan. 24, 1978.
9. A di~ital block-processor for SSB-FDM modulation and demodulation; P.M. Terrell, P.J.W. Rayner; IEEE
transactions on communications, vol. COM-23, No. 2, February 1975, pages 282 - 286.
10. Arrangement for processing auxiliary signals in a frequency division-multiplex transmission system;
Canadian Patent 1,075,837 which issued to Telecommunic-ations Radioelectriques et Telephoniques T.R.T. on April 15, 1980.
.
~114534 11. Digital Signal Processing; A.V. Oppenheim, R.W. Schafer; Prentice-Hall, inc., Englewood Cliffs, -New Jersey 1975.
12. A digital signal processing approach to inter-polation; R.W. Schafer, L.R. Rabiner: Proceedings of the IEEE, Vol. 61, No. 6, June 1973, pages 692 - 702.
13. Digital Signal Processing; B. Gola~ C.M. Rader;
McGraw-Hill Book Company, 1969.
14. Theory and Application of Digital Signal Pro-cessing; L.R. Rabiner, B. Gold; Prentice-Hall, i inc., Englewood Cliffs, New Jersey 1975.
15. Digital filter; Canadian Patent 1,011,823 which issued to Telecommunications Radioelectriques et 15 Telephoniques T.R.T. on June 7, 1977.
16. Interpolating digital filter; our Canadian Patent 1,039,364 which issued on September 26, 1978.
17. Interpolating non-recursive digital filter;
our Canadian Patent 1,053,762 which issued on May 1, 1979.
18. System for the transmission of analog signals by means of pulse code modulation; Canadian Patent 950,971 which issued to Telecommunications Radio-electriques et Telephoniques T.R.T. on July 9, 1974. ~;
E. Descri tion of the embodiments P :.
E(l) Introduction E(l.l) Digital signals and their frequency spectra As is noted already that a digital signalis a signal which is both time and amplitude dis-~$~45~4 10.6.77 '-crete. Such a signal may, for example, be obtained by sampling an analog signal b(t) at instants n.Tb wherein n = O, + 1, * 2,.... and wherein Tb ~epre-sents the sampling period. The samples of the sig~
nal b(t) thus obtained are each quantized and may be converted into a multi-digit digital number.
The nth sample of the analog signal b(t) will be denoted by b(n) and'will be called component.
Now the digital signal can formally be represent~
ed by the series fb(n)}
Digital signals and, in general, discrete signals are usually graphically represented in the manner shown in Eigure 1. (see also reference 11).
., Although the abscissa is drawn as a continuous line it should be noted that b(n) is only defined ; .
for integer values of n. It is incorrect to think that b(n) is zero "for all values of n which are not an integer; b(n) is simply undefined for non-integer values of n.
ZO The frequency spectrum of this discrete signal ¦ b(n)} is given by the equation oo ,~, ~( ~ ) = ~ b(n) e b (4) Eq. (4) shows the frequency spectrum B( ~ ) is periodical and its period is equal to b so that:
_ 26 - ' 10.6.77 -",' :.' ''; ~
B(~ ~ ~ T ) = B(hJ) where CX is an integral (5) : .
If b(n) represents a real signal it f.urther-more follows from Eq, (4) that:
B~(~) = B(2T _ ~)) (6) ~ ~ In Eq. (6) B (hJ) represents the complex conjugated value of B(IJ), By means of the inverse transform the ~ ~
digital signal { b(n)} can be obtained from its . .
: frequency spectrum B( W). This inverse transforma-: , tion is mathematically given by the equation.
2 ~ : . :
b(n) = 2b~ ~ B(~ ) e b d W (7) .
~ ~ o .~:
: Because B(~J) is a periodic funotion in ~(see (5)), ~:~ any arbitrary frequency ~interval of the length r,.
T may be taken as the integration frequency in-.terval. Consequently the description of this fre-quency spectrum can be limited to the description ~ :
of one frequency interval. This frequency interval .
will be indicated by QB:and will be called the fun- ~-damental interval. In what follows hereinafter the : ~
frequellcy spectrum of a-digital signal will be .
desoribed i.n the fundamental interval 0 ~ W C Q . ~ ~;
Figure 2 shows diagra~mmatically the f~equency ., .
- ' ' P~IN 8731 ~ S ~ ~ ~ 10.6.77 -spectrum of the digital signal{b(n ~ shown in Fi-gure 1. ;
~, E(1.2) Samplin~ rate alteration In the arrangements to be described ele-::
5 - ments are used for increasing or reducing the sampling rate which is associated with the digital input sig-nal of this element. An elément which is used for :-. :.
increasing the sampling rate will be indicated by the symbol shown in Figure 3 and will be called ~ -SRI-element (SRI = Sample Rate Increase~. In the -, ~ . . . .
symbol of Figure 3 q represents the increase fac- - -~tor and q is an integer. If, more in particular, ;~
~ ~ the sampling rate which is associated with the - `
- m digital input signal of this SRI-element is equal to 1 then the sampling rate which is associated with its digital output signal is equal to -T.
The operation of an SRI-element is as follows.
Between each bwo sucoessi~e components o~ the digital input signal, q-1 zero-valued components are inserted. If, for example the digital signal fb(n)~ shown in Figure 1 is applied to the SRI-element of Figure 3, a digital output signal fd(n)~ is obtained which in case q = 3 has the form depicted in Figure 4. The operation of this --SRI-element is mathematically described by the expressions~
:~' -; - - 28 P~IN 8731 10,6.77 ~...
d(n) = b(q) for: n~0, +q, +2q~,,. (8j -= 0 for: all other values of n.~ ~
,. :
Because the sampling rate with which the components d(n) occur is equal to T the fundamental interval of the frequency spectrum D(~J) of the signal ~, ~d(n)~ is equal to: `
From (8) and (4) it follows that:
D( W ) = B(~ (10) Because ~ D ~ q ~B the fundamental interval of the frequency spectrum D( ~) consequently comprises q fundamental intervals of~the frequency spectrum B( W3. For q = 3 this is diagrammatically depict-ed in Figure 5. ' An element which i9 used for reducing the sampling rate will be indicated by the symbol shown in Fi-:: ' ; . ' -gure 6 and will be called SRR-element (SRR = Sample -Rate Reduction). In the symbol of Figure 6 q repre~
sents the reduction factor and q is an integer. If more in particular the~ sampling rate which is as~
,~ . .
~ sociated with the digital input signal of this -:;:
SRR-element is equal to T~ then the sampling rate ~ -which is associated with its digital output sig-nal to 1T. Such a SRR-element operates as follows.
: .' 10.6.77 9L ~ -Each time one out of each q successive input samples .
of this element is selected and applied to its out-put. If the digital input sig~al of the SRR-element :
is formed by the signal ~c(n)~ deyicted in Figure .
7, a digital signal fe(n)~ is obtained at its in- .
put whioh in case q = 3 has the form depicted in ~, Figure 8, The operation of this SRR-element can ~ :
mathematically be described by the expression . e(n) = c(nq) ( 11 ) .
If now a sampling rate T is associated with fc(n)3 ~ :
and a sampling rate - with ~e(n~ then it holds . .
: that Te = qTc. The fundamental interval of the fre-quency spectrum C(~J) of {c(n)} is now equal to .::~ :
a = ~ 7r and the fundamental interval of the rrequency spectrum E( W ) of {e(n)} is equal to .:::.
E = T = T~ so that ~ E = qQc- By substituting ';~
expression (11 ) in expression (4) it can be proved that .
the relation bebween E(~ ) and C(W ) can mathematical- .~
ly be described.by the expression: .
q ~1 C r~ + (k-1) 2~ ] (12) The above is diagrammatically depicted in the Figures 9 and 10. 11-The SRI-element shown symbolically in Fi- -gure 3 and the SRR-element shown symbolically in .
Figure 6 will not be found as a concrete circuit ; ~:
I .
10.6.77 , -~
in the practical embodiments of the arrangements to be described but in combination with other elements.
These SRI- and SRR-elements must be considered as mathematical elements which are exclusively used ~or simplifying the description of the operation of the various embodiments and for a better understand-. .. .
ing thereof. For these reasons no practical circuit of these elements will be given here, E(1.~) Sideband lrterohan~i,n~ modulator. I `
The arrangements to be described also utilize an element for interchanging the upper and ¦ , lower sideband in the frequency spectrum of the digital signal applied bo this element. This element will be represented by the symbol depicted~;in Figure 11 and ~, t5 will be called sideband interchanging modulator.
This sideband interchanging modulator operates as ~;
follows. Each time one out o~ two components of the input componento f(n) o~ the modulator is multiplied by a factor-1. If more in particular the components f(n) depicted in Figure 2 are applied to this mo-dulator, the output components g(n) depicted in Figure 13 are~ obtained. The operation of this side-band interchanging modulator can be described ~"
mathematically by the expression: -~
g(n) = (-1) ~(n) (13) By substituting expression (13) in (4) it can be proved that the relation between the frequency `: :`.
'`' PI~N 8731 10.6.7'7 i~ S3~
spectrum F( W ) of ~ f(n)~ and the frequency spectrum G(~J ) of {g(n)~ can be described mathematically by ~ -the expression: :
G(hJ) = F ~ 2i-1) T ~ (14) *
' . - ..... ;
wherein T represents the sampling rate associat-ed with both ff(n)~ and fg(n)~ All this is .
diagrammatically depicted in Figures 14 and 15. : ,:
A possible embodiment of such a sideband ;. ~- .
interchanging modulator is shown in Figure 11a. It ~10 comprises two AND-gates 11(1) and 11(2), an OR-gate
; coupled devices" (CCD's) are, for example, used. -~
Arrangements which are suitable for pro-~- cessing digital signals are called digital arrange-ments whilst arrangements which are suitable for processing sampled data signals are indicated cor-respondingly as sampled data arrangements.
What follow~ hereinafter applies to both a digital and to a sampled data arrangement and the invention will be further explained with reference to a digital TDM-FDM arrangement and a digital FDM-TDM arrangement. -;
C. Short description of the Fi~ures.
Figure 1 shows the grafical representa- - -tion of a digital signal Figure 2 shows schematically the fre-quency spectrum of a digital si~nal; -:
-~ ;
10.6.7~
i34 . ~
' ~:
`. ;.
Figure 3 shows the symbol for a device for increasing the sampling rate (SRI-element);
Figures 4 and 5 show diagrams for explain-ing the operation of the SRI-element;
Figure 6 shows the symbol of a device for reducing the sampling rate ( ~ -element) and the figures 7, 8, 9 and 10 show some diagrams for ex-plaining the operation of the SRR-element; :.
Figure 11 shows the symbol of a sideband~
interchanging modulator; ~.
Figure 11a shows an embodiment of a side~
band interohanging modulator; .
The Figures 12, 13, 14 15 show some dia- ~
.
grams for explaining the operation of the sideband ;~.
interchanging modulator;
Figure 16 shows the symbol for a oomplex modulator and .
Figure 17 shows an embodiment of such a .. ~
.. .
modulator; whilst the Figures 18a, 18b and 18c show some frequency spectra of the signals pro-duced by the complex modulator;
Figure 19 shows the frequency spectrum ~of a real input signal of the TDM-FDM arrangement;
Figures 20a and 20b show two possible fre-quency spectra of the real SSB-FDM signal;
Figures 21 shows the bacic diagram of the TDM-FDM arrangement according to the invention; ` ~ :
. ~
,. '~
Figure 22, which is on the same sheet as ` -Fig. 20a and 20b, shows the frequency spectra of some ;-input signals of the transformer in the TDM-FDM
arrangement according to the invention;
Figure 23 shows an embodiment of a two-point transformer for real input signals and complex multiplication factors and Figure 25, on the same sheet, shows the symbol of such a transformer;
Figure 24 shows an embodiment of a two-point transformer for complex input signals and complex multiplication factors and Figure 26, which is on the ;~
same sheet as Fig. 23 and Fig. 25, shows the symbol of such a transformer; -Figure 27 shows an eight-point fast trans-former for use in the TDM-FDM arrangement shown in Figure 21;
Figure 28 shows the general set-up of a signal channel of the device according to Figure 21 and the Figures 28a, 28b and 28c show modified versions of the signal channel of Figure 28;
Figures 29 and 30 show modified versions of -the TDM-FDM arrangement shown in Figure 21;
Figure 31, which is on the same sheet as Fig. 29, shows a single circuit configuration which is used for the set-up of the signal channels in the arrangement according to Figure 30;
The Figures 32, 33, 35 and 38 and 40 show -some modifications of the circuit configuration shown in Figure 31;
.
: ~:
PI~N 8731 10.6.77 The Figures 34, 36, 37, 41, 42 and 43 show -.
some transfer functi.ons of filter means which are used in the circuit configurations of the Figures :
32, 33, 35, 38 and 40;
S ~igure 39 shows a detailed embodiment of .~
a TDM-FDM arrangement wherein the transformer is -.- ;
based on a Hadamard matrix and wherein real sig-nals are applied to the transformer; ~ .
. Figure 44 shows an embodiment of a TDM-~DM; .
arrangement wherein the transformer is based on a complex matrix and to which complex signals are .
applied;
The Figures 45 and 46 show some transfer functions of the digital filters used in the de-vice of Figure 44;
Figure 47 shows an embodiment of an FDM-TDM arrangement.
D. References 1. A third method of generation and detection of single-sideband signals; D.K. Weaver~ Proceedings of the IRE, December 1956, pages 1703 - 1705.
2. Terminology in digital signal processing;
L.R. Rabiner c.s.; IEEE transactions on audio : .
and electroacoustics, vol. AU-20, No. 5, December 1972, pages 322 - 337. .
. On digital single-sideband modulators; S~.
Darlington; IEEE transaotions on circuit theory; .
- 23 - ~
.
~ 534 ~HN ~731 vol. CT-17, No. 3, August 1970, pages 409 - 414.
4. Design of digital filters for an all digital fre-quency-division multiplex time-division multiplex translator; S.L. Freeny c.s.; IEEE transactions on circuit theory, vol. CT-18, No. 6, November 1971, pages 702 - 710.
5. SSB/FDM utilizing TDM digital filters; C.F. Kurth~
IEEE transactions on communication technology, vol.
COM-l9, No. 1, February 1971, pages 63 - 71.
6. TDM-FDM transmultiplexer; digital polyphase and FFT; M.G. Bellanger, J.L. Daguet; IEEE transactions on communications, vol. COM-22, No. 9, September 1974, pages 1199-1205. ~ -7. Single-sideband system for digital processing ~ `
15 of a plurality of channel signals; Canadian Patent -987,742 which issued to Telecommunications Radio-electriques et Telephoniques T.R.T. on April 20, 1976. -8. Circuit arrangement for digitally processing a given number of channel signals; Canadian Patent 1,025,135 which issued to Telecommunications Radio-electriques et Telephoniques T.R.T. on Jan. 24, 1978.
9. A di~ital block-processor for SSB-FDM modulation and demodulation; P.M. Terrell, P.J.W. Rayner; IEEE
transactions on communications, vol. COM-23, No. 2, February 1975, pages 282 - 286.
10. Arrangement for processing auxiliary signals in a frequency division-multiplex transmission system;
Canadian Patent 1,075,837 which issued to Telecommunic-ations Radioelectriques et Telephoniques T.R.T. on April 15, 1980.
.
~114534 11. Digital Signal Processing; A.V. Oppenheim, R.W. Schafer; Prentice-Hall, inc., Englewood Cliffs, -New Jersey 1975.
12. A digital signal processing approach to inter-polation; R.W. Schafer, L.R. Rabiner: Proceedings of the IEEE, Vol. 61, No. 6, June 1973, pages 692 - 702.
13. Digital Signal Processing; B. Gola~ C.M. Rader;
McGraw-Hill Book Company, 1969.
14. Theory and Application of Digital Signal Pro-cessing; L.R. Rabiner, B. Gold; Prentice-Hall, i inc., Englewood Cliffs, New Jersey 1975.
15. Digital filter; Canadian Patent 1,011,823 which issued to Telecommunications Radioelectriques et 15 Telephoniques T.R.T. on June 7, 1977.
16. Interpolating digital filter; our Canadian Patent 1,039,364 which issued on September 26, 1978.
17. Interpolating non-recursive digital filter;
our Canadian Patent 1,053,762 which issued on May 1, 1979.
18. System for the transmission of analog signals by means of pulse code modulation; Canadian Patent 950,971 which issued to Telecommunications Radio-electriques et Telephoniques T.R.T. on July 9, 1974. ~;
E. Descri tion of the embodiments P :.
E(l) Introduction E(l.l) Digital signals and their frequency spectra As is noted already that a digital signalis a signal which is both time and amplitude dis-~$~45~4 10.6.77 '-crete. Such a signal may, for example, be obtained by sampling an analog signal b(t) at instants n.Tb wherein n = O, + 1, * 2,.... and wherein Tb ~epre-sents the sampling period. The samples of the sig~
nal b(t) thus obtained are each quantized and may be converted into a multi-digit digital number.
The nth sample of the analog signal b(t) will be denoted by b(n) and'will be called component.
Now the digital signal can formally be represent~
ed by the series fb(n)}
Digital signals and, in general, discrete signals are usually graphically represented in the manner shown in Eigure 1. (see also reference 11).
., Although the abscissa is drawn as a continuous line it should be noted that b(n) is only defined ; .
for integer values of n. It is incorrect to think that b(n) is zero "for all values of n which are not an integer; b(n) is simply undefined for non-integer values of n.
ZO The frequency spectrum of this discrete signal ¦ b(n)} is given by the equation oo ,~, ~( ~ ) = ~ b(n) e b (4) Eq. (4) shows the frequency spectrum B( ~ ) is periodical and its period is equal to b so that:
_ 26 - ' 10.6.77 -",' :.' ''; ~
B(~ ~ ~ T ) = B(hJ) where CX is an integral (5) : .
If b(n) represents a real signal it f.urther-more follows from Eq, (4) that:
B~(~) = B(2T _ ~)) (6) ~ ~ In Eq. (6) B (hJ) represents the complex conjugated value of B(IJ), By means of the inverse transform the ~ ~
digital signal { b(n)} can be obtained from its . .
: frequency spectrum B( W). This inverse transforma-: , tion is mathematically given by the equation.
2 ~ : . :
b(n) = 2b~ ~ B(~ ) e b d W (7) .
~ ~ o .~:
: Because B(~J) is a periodic funotion in ~(see (5)), ~:~ any arbitrary frequency ~interval of the length r,.
T may be taken as the integration frequency in-.terval. Consequently the description of this fre-quency spectrum can be limited to the description ~ :
of one frequency interval. This frequency interval .
will be indicated by QB:and will be called the fun- ~-damental interval. In what follows hereinafter the : ~
frequellcy spectrum of a-digital signal will be .
desoribed i.n the fundamental interval 0 ~ W C Q . ~ ~;
Figure 2 shows diagra~mmatically the f~equency ., .
- ' ' P~IN 8731 ~ S ~ ~ ~ 10.6.77 -spectrum of the digital signal{b(n ~ shown in Fi-gure 1. ;
~, E(1.2) Samplin~ rate alteration In the arrangements to be described ele-::
5 - ments are used for increasing or reducing the sampling rate which is associated with the digital input sig-nal of this element. An elément which is used for :-. :.
increasing the sampling rate will be indicated by the symbol shown in Figure 3 and will be called ~ -SRI-element (SRI = Sample Rate Increase~. In the -, ~ . . . .
symbol of Figure 3 q represents the increase fac- - -~tor and q is an integer. If, more in particular, ;~
~ ~ the sampling rate which is associated with the - `
- m digital input signal of this SRI-element is equal to 1 then the sampling rate which is associated with its digital output signal is equal to -T.
The operation of an SRI-element is as follows.
Between each bwo sucoessi~e components o~ the digital input signal, q-1 zero-valued components are inserted. If, for example the digital signal fb(n)~ shown in Figure 1 is applied to the SRI-element of Figure 3, a digital output signal fd(n)~ is obtained which in case q = 3 has the form depicted in Figure 4. The operation of this --SRI-element is mathematically described by the expressions~
:~' -; - - 28 P~IN 8731 10,6.77 ~...
d(n) = b(q) for: n~0, +q, +2q~,,. (8j -= 0 for: all other values of n.~ ~
,. :
Because the sampling rate with which the components d(n) occur is equal to T the fundamental interval of the frequency spectrum D(~J) of the signal ~, ~d(n)~ is equal to: `
From (8) and (4) it follows that:
D( W ) = B(~ (10) Because ~ D ~ q ~B the fundamental interval of the frequency spectrum D( ~) consequently comprises q fundamental intervals of~the frequency spectrum B( W3. For q = 3 this is diagrammatically depict-ed in Figure 5. ' An element which i9 used for reducing the sampling rate will be indicated by the symbol shown in Fi-:: ' ; . ' -gure 6 and will be called SRR-element (SRR = Sample -Rate Reduction). In the symbol of Figure 6 q repre~
sents the reduction factor and q is an integer. If more in particular the~ sampling rate which is as~
,~ . .
~ sociated with the digital input signal of this -:;:
SRR-element is equal to T~ then the sampling rate ~ -which is associated with its digital output sig-nal to 1T. Such a SRR-element operates as follows.
: .' 10.6.77 9L ~ -Each time one out of each q successive input samples .
of this element is selected and applied to its out-put. If the digital input sig~al of the SRR-element :
is formed by the signal ~c(n)~ deyicted in Figure .
7, a digital signal fe(n)~ is obtained at its in- .
put whioh in case q = 3 has the form depicted in ~, Figure 8, The operation of this SRR-element can ~ :
mathematically be described by the expression . e(n) = c(nq) ( 11 ) .
If now a sampling rate T is associated with fc(n)3 ~ :
and a sampling rate - with ~e(n~ then it holds . .
: that Te = qTc. The fundamental interval of the fre-quency spectrum C(~J) of {c(n)} is now equal to .::~ :
a = ~ 7r and the fundamental interval of the rrequency spectrum E( W ) of {e(n)} is equal to .:::.
E = T = T~ so that ~ E = qQc- By substituting ';~
expression (11 ) in expression (4) it can be proved that .
the relation bebween E(~ ) and C(W ) can mathematical- .~
ly be described.by the expression: .
q ~1 C r~ + (k-1) 2~ ] (12) The above is diagrammatically depicted in the Figures 9 and 10. 11-The SRI-element shown symbolically in Fi- -gure 3 and the SRR-element shown symbolically in .
Figure 6 will not be found as a concrete circuit ; ~:
I .
10.6.77 , -~
in the practical embodiments of the arrangements to be described but in combination with other elements.
These SRI- and SRR-elements must be considered as mathematical elements which are exclusively used ~or simplifying the description of the operation of the various embodiments and for a better understand-. .. .
ing thereof. For these reasons no practical circuit of these elements will be given here, E(1.~) Sideband lrterohan~i,n~ modulator. I `
The arrangements to be described also utilize an element for interchanging the upper and ¦ , lower sideband in the frequency spectrum of the digital signal applied bo this element. This element will be represented by the symbol depicted~;in Figure 11 and ~, t5 will be called sideband interchanging modulator.
This sideband interchanging modulator operates as ~;
follows. Each time one out o~ two components of the input componento f(n) o~ the modulator is multiplied by a factor-1. If more in particular the components f(n) depicted in Figure 2 are applied to this mo-dulator, the output components g(n) depicted in Figure 13 are~ obtained. The operation of this side-band interchanging modulator can be described ~"
mathematically by the expression: -~
g(n) = (-1) ~(n) (13) By substituting expression (13) in (4) it can be proved that the relation between the frequency `: :`.
'`' PI~N 8731 10.6.7'7 i~ S3~
spectrum F( W ) of ~ f(n)~ and the frequency spectrum G(~J ) of {g(n)~ can be described mathematically by ~ -the expression: :
G(hJ) = F ~ 2i-1) T ~ (14) *
' . - ..... ;
wherein T represents the sampling rate associat-ed with both ff(n)~ and fg(n)~ All this is .
diagrammatically depicted in Figures 14 and 15. : ,:
A possible embodiment of such a sideband ;. ~- .
interchanging modulator is shown in Figure 11a. It ~10 comprises two AND-gates 11(1) and 11(2), an OR-gate
3), a multiplier 11(4), a modulo-2-counter 11(5) '~
: to which clock pulses are.. applied which are derived from a clock pulse generator 11(6). A decoding net-;~ work 11(7) having two outputs is ccnnected to the : 15 modulo-2-adder, Furthermore the oomponents f(n) which occur at a frequency 1/Tf are applied to both .
~ . AND-gates~and the clock pulse frequency of said : ~;
~ :
... clock pulses is also equal to 1/Tf. In response to ; a first of two successive clock pulses the AND-gate -::
. 20 11(1) is made conductive and the AND-gate 11(2) is ~ -.. .
.. cut off. In response to the second clock pulse of :
the second successive clock pulse the AND-gate 11(1) is cut off and the AND-gate 11(2) is made .
conductive, The output components f(n) of AND-gate 25 . 11(2) are multiplied by a ~actor -1 in the multiplier . - -11(4). ~ ~
!
32 ::
.
PHN 8731 ., i~4~4 10.6.77 E(1.4) The complex modulator, 'Besides said~sideband interchanging modu-l-ator in the arrangements'to be described also an - element can be used for converting a real digital ~--signal into a complex digital signal. This element ,~
can be represented by the symbol depicted in Figure 16 and will be called complex modulator.:In this . ,' complex modulator the compo- " ~
nents f(n) of the digital input signal, occurring at ,, ; ,, a rate 1/TX are each multiplied by a factor e j 1 Tf, ` ~, to produce the complex digital output signal ,' ~ p(n) ~ f(n)cos(W 1nTf)+jf(n)sin~J1nTf). This com- ~, ,~ 'plex signal comprises a real part Re ~ p(n)~ and and an imaginary parb Im [p(n)~ , wherein~
Re ~p(n)~ = f(n)cos(~J~nTf) ,-.
Im [p(n)] - f(n)Sin( W1nTf)-In a practical embodiment of such a modulator the components ~e t p(n)~ and Im [ p(n)~ are produced at separate outputs of the modulator. In practice this complex modulator can be constituted by that part of the digital Weaver modulator (see references 3, 4 and 5) which is depicted in Figure 17 and whi¢h -does not' need any further explanation. For complete-ness~ sake Figure 18a shows symbolically some p'eriods of the frequency spectrum P(~/) of the complex digi- -~' ta~ signa] { p(n)~ if the digital s~nal ~ f(n)~ ~ ~
., '~
- ~
: ;
10.6.77 :
1~1453 , having the frequency spectrum depicted in Fig. 14 is ~
applied to this complex modulator. This frequency ~ ;~r - spectrum P(~U) is mathematically given by thé ex- ~ , ....
pression:
: ~5 P(~J) = F(~J ~ ~J1) (15) ~:
Figures 18b and 18c respectively show the frequency - :
spectra p(1)(~) and P(2)( W ) associated with the :
signals Re ~ p(n)~ and Im [ p(n)~ produced by the ¦;
arrangement depicted in ~igure 17. These frequency-~ spectra ¢an mathematically be expressed by: . :
~ F(~ + ~1) ~:~
: .
) 2j F(~ _ W ) 1 F ( .
~ (16) :;
; . E(2~ The_TDM-FDM-arranFement. ~ :
E(2 1~ Introdu_tion .~- :
The digital TDM-FDM arrangement is an ar- , ;
rangement for con~erting N r~al baseband signals :::
~ xk(n)~ , with k = 1, 2, 3, ..... N, into a real .
: digital baseband single-sideband frequency division-multiplex signal fy(n)~ . Let us suppose that the -sampling period which is associated with each of N
signals f xk(n~ is equal to T. The frequellcy spec-trum Xk(W ) of ~xk(n)~- is diagrammatically depict-ed in Figure 19 and has a fundamental inter~al .
_ 34 _ ~ :
.
''~ ' PHN ~731 10.6.77
: to which clock pulses are.. applied which are derived from a clock pulse generator 11(6). A decoding net-;~ work 11(7) having two outputs is ccnnected to the : 15 modulo-2-adder, Furthermore the oomponents f(n) which occur at a frequency 1/Tf are applied to both .
~ . AND-gates~and the clock pulse frequency of said : ~;
~ :
... clock pulses is also equal to 1/Tf. In response to ; a first of two successive clock pulses the AND-gate -::
. 20 11(1) is made conductive and the AND-gate 11(2) is ~ -.. .
.. cut off. In response to the second clock pulse of :
the second successive clock pulse the AND-gate 11(1) is cut off and the AND-gate 11(2) is made .
conductive, The output components f(n) of AND-gate 25 . 11(2) are multiplied by a ~actor -1 in the multiplier . - -11(4). ~ ~
!
32 ::
.
PHN 8731 ., i~4~4 10.6.77 E(1.4) The complex modulator, 'Besides said~sideband interchanging modu-l-ator in the arrangements'to be described also an - element can be used for converting a real digital ~--signal into a complex digital signal. This element ,~
can be represented by the symbol depicted in Figure 16 and will be called complex modulator.:In this . ,' complex modulator the compo- " ~
nents f(n) of the digital input signal, occurring at ,, ; ,, a rate 1/TX are each multiplied by a factor e j 1 Tf, ` ~, to produce the complex digital output signal ,' ~ p(n) ~ f(n)cos(W 1nTf)+jf(n)sin~J1nTf). This com- ~, ,~ 'plex signal comprises a real part Re ~ p(n)~ and and an imaginary parb Im [p(n)~ , wherein~
Re ~p(n)~ = f(n)cos(~J~nTf) ,-.
Im [p(n)] - f(n)Sin( W1nTf)-In a practical embodiment of such a modulator the components ~e t p(n)~ and Im [ p(n)~ are produced at separate outputs of the modulator. In practice this complex modulator can be constituted by that part of the digital Weaver modulator (see references 3, 4 and 5) which is depicted in Figure 17 and whi¢h -does not' need any further explanation. For complete-ness~ sake Figure 18a shows symbolically some p'eriods of the frequency spectrum P(~/) of the complex digi- -~' ta~ signa] { p(n)~ if the digital s~nal ~ f(n)~ ~ ~
., '~
- ~
: ;
10.6.77 :
1~1453 , having the frequency spectrum depicted in Fig. 14 is ~
applied to this complex modulator. This frequency ~ ;~r - spectrum P(~U) is mathematically given by thé ex- ~ , ....
pression:
: ~5 P(~J) = F(~J ~ ~J1) (15) ~:
Figures 18b and 18c respectively show the frequency - :
spectra p(1)(~) and P(2)( W ) associated with the :
signals Re ~ p(n)~ and Im [ p(n)~ produced by the ¦;
arrangement depicted in ~igure 17. These frequency-~ spectra ¢an mathematically be expressed by: . :
~ F(~ + ~1) ~:~
: .
) 2j F(~ _ W ) 1 F ( .
~ (16) :;
; . E(2~ The_TDM-FDM-arranFement. ~ :
E(2 1~ Introdu_tion .~- :
The digital TDM-FDM arrangement is an ar- , ;
rangement for con~erting N r~al baseband signals :::
~ xk(n)~ , with k = 1, 2, 3, ..... N, into a real .
: digital baseband single-sideband frequency division-multiplex signal fy(n)~ . Let us suppose that the -sampling period which is associated with each of N
signals f xk(n~ is equal to T. The frequellcy spec-trum Xk(W ) of ~xk(n)~- is diagrammatically depict-ed in Figure 19 and has a fundamental inter~al .
_ 34 _ ~ :
.
''~ ' PHN ~731 10.6.77
4~34 ~
n X = ~ As ~ Xk(n)~ is a real signal, Xk(~
satisfies the relation:
k( T ) = Xk~~ for ~ ~J 0 < T (17~ :, Suppose that the sampling rate associated ~ -with the desired real FDM signal ~ y(n)~ is equal to - , and that:this sampling rate is an integral .
multiple of T. So assume: T = MT, wherein M ~ N
' and wherein M is an integer. The fundamental inter- :
. val Qy of the frequency spectrum of thi-s FDM-'sig-nal therefor is equal to ~ y = T = Mnx. As .
y(n)~ must be a real signal the frequency spec- ¦
trum Y( W) of this FDM-signal must satisfy the re- :
-h~). Therefor~ this fre-'- ,:
quency spectrum must have in general the form which is diagrammatioally depicted in Figure 20a for N = 4 and M = 5, The channel-signals are each located in a aubband Fk of length T wherein k = 1, 2, 3; N.
Eaçh subband is characterized by the frequency range :
J, + (k-1)~ T~ ~ W < W1 + k.T . In what fo lows here-inafter it will be assumed that 0 h~1 < T . As Y(~J) must represent the frequency spectrum of a FDM-signal which comprises a frequency multiplexr of N baseband signals ~ x(n)~ , the following ex- ,:
pressions must be satisfied: .
~ !
10.6.77 3~ `*
¦ I O ( ) T ~ = Xk(hJo) (18~) and~
[ T - { 1 ~ ~J o + ( k-1) T~
= y~ `[Wl + /~J + (k-l ) ~ l = Xk ~ W O) = Xk ( T o , (18b)l ~ ;
: ~, ~
~: ~ It should be noted that for ~1 = the ~ .:
value of M may be taken equal to N so: M = N,` so . :
~: that the frequency spectrum:Y(~J) has the form de- ~
; .:' . picted in Figure ZOb. .
E(2.2) General implementation of the TDM-FDM-arran~ement :
Figure 21 shows diagrammatically the imple- ~ ;
mentation of a digital TDM-FDM arrangement for con-verting N real digital baseband signals ~xk(n)3 , (k = 1, 2, 3~ ... N; n = .... -2, -1, O, ~ 2~ ...... ), - :
ha~ing associated therewith a sampling period T, into a real digital baseband single-sideband frequency-division multlplex signal ~y(n)~ having associat-ed therewith a sampling period Ty = M. This arrange- ¦
ment comprises an input circuit formed by N input channels 1(1), 1(2), 1(33, .. ~ 1(N). A digital baseband signal { xk(n)~ is applied to each of these input channels. The frequency spectra of these .
. ~ .
P~IN 8731 10.6-77 '~
signals follow from (4) and these spectra are shown ~':
in Figure 19. For generating an FDM-signal having .
the frequency spectrum depicted in Figure 20a for - N = 4 a comp].ex modulator 1(1,1), 1(1,2), .. 1(1,N) ::
is included in each of the input channels and the - ~
input channels having an even number are provided ~ . -with a sideband interchanging modulator 2(1); 2(2); ~ :
.2(3); ... 2(N/2). .The input channels are connected . ,.
to inputs of a transformation device 3. The digital ~ :
signals generated by the input channels and being ~: applied to the inputs of the transformation device .~ -3 will be indicated by frk(n)~ . The relation- -~ .
:-:
~: : ship between the components rk(n) and xk(n) can mnthematically be given by the expression:
rk(n) = xk(n) ei 1 for k is odd k(n) = (-1) xk(n) e~ ~1nT for k is even .~
~, :';, (19) ~ -The frequency spectrum Rk(W ) of the digital signal - :
~rk(n)~ can be derived f~om (14) and (15), as well -.
as from the Figures 14, 15 and 18a and is given by: .
k(~J) = Xk ~ 4 - ~1 ~ (k-l). TW ~ (20) For k is odd it applies that k-1 is even so that k(~J) = Xk(h~ ~ ~J1) for k = odd.
' ~" ~ pl~ 8731 ~ 534 10.6.77 ~ ~
' ` . ~"''' ' ' '' For k is even it applies that ~-1 is odd so that -Rk(W ) = ~ ~ -~J1 - (k-1). T~ ~ for k = even ~ [ 1 T
The spectrum defined by (20) is depicted in Figure 22.
~5 ~ The transformation device 3 produces N di-gital signals ~ sm(n)~ , with m = 1, 2, 3, .,,N, having associated therewith a sampling period T, These digital signals f sm(n)~ are each applied to a signal channel 4(t), 4(2),... 4(N). The proces- :
~l0 sing operation which is performed by the transfor-~ mation device 3 can mathematically be described by ; ~ the expression: ;~
, N
(n) = ~ amk rk( ) (21) ~ ;' ~ ~
~; m 1, 2, 3~ .,.N.
15~ In this expression a k represents a multiplication factor of constant value. This multiplication fac-tor can be a real but also a complex number. If in general it is assumed that amk is a complex number which can be given by: ;~ ~-20~m~ ~ mk i ~ mk (22) then sm(n) also represents a complex number.
. ,~
:~ 38 _ .'' .
:;.
10.6.77 ,~
''~ '~' The general implementation of the transfor- -mer 3 as well as the general implementation of the signal channels 4(m) will be described in chapters E(2.4) and E(2,5). At this moment it is supposed ,:
that the complex signals ~s (n)~ are available in a form which is suitable for further proces-,,: ~..:
sing.
From (213 it is clear that each compo~ ;
~ nent sm(n) i9 formed by a linear combination of com- . ~ ' ponents rk(n). As a result of the linear character `
;
~: of (21) it applies that the frequency spectrum `
m(~J) f~ {Sm(n)~ is given by~
Sm( ~ ) = ~ amk k(W ) (23) The signal channels 4(1), 4(2),;..4(N) each comprise a cascade arrangement of an SRI-ele-ment 5(1), 5(2),...5(N) and a digital filter 6(1), 6(2),.,.6(N) and are connected to inputs of an adder ~' 7, In Figure 21 the components of the OUtpllt signal of the SRI-element 5(m) are indicated by tm(n);
the componentsof the output signal of the digital filter 6(m) are indicated by um(n); and the compo-.
nents of the output signal of the adder 7 are in- -; dicated by v(n). As ~v(n)~ in general represcnts a complex digital signal and as we are only in-` terested in a real digital output signal y(n) ": .
_ 39 , PHN 8731 10.6,77 having the frequency spectrum Y(~ ) which for N = 4 ..
is depicted in Figure ZOa, the components v(n) are ~ .
applied to a selec~-or 8 which only produces the real portion of the complex component v(n) at its output.
In order to find a mathematical expression, for the output signal ~ y(n)~ it should be noted ',:-that the relation between t (n) and sm(n) is given by expression (8). From (10) it follows that the .
, frequency spectrum Tm( W) of ~tm(n)~ is given by:
T (~) = S ( ~ ) : : (24) : T = M Q s = T .. . -M being an integer and M~ N :~ .`.
The digital signal f tm(n)~ is filtered ~: ~ in the filter 6(m). If the transfer function of the :
filter 6(m).is represented by Hm( W ) and the fre-:15 quency ~pectrum of the signal ~ um(n)~ by Um(~
: then it applies that:
.: : .
m(~J) = Hm(~ )Tm( ' (25) m = 1, 2, 3, ...N
The.complex output signal ~ v(n)~ of this TDM-FDM arrangement is now obtained by adding the signals f u~(n)~ together so that:
N :.~
v(n) = ~ um(n) (26) .:
P~IN 8731 ' 10.6.77 and consequently: ~-- N ~ ;
y(n) = Re L ~1 Um(~ (27) : .
The frequency spectrum V( W) of ~ v(n)~
is given by:
.:
N
V( W) = ~ Um(~J) (28) ,~
;
As (27) can be written in the form:
y(n) = Re ~v(n)~ = ~ [v(n)~v (n)~ (~9) wherein v (n') represents the complex conjugate of v(n) and as the frequency spectrum of {v(n)~is equal to V( W ), so that the frequency spectrum of { vX(n)~is equal to V~( T ~ W ) it holds that:
Y :
( j [v(~J) + V ( T ~~ 3)~
E(2.~) The FDM- conditions In order to generate an SSB-FDM-signal having the frequency spectrum Y(W ), which for N = 4 I .:
is depicted in Figure 20a the transfer function H~n( W) --of the filter 6(m) must satisfy a very special con-dition. This condition will be called "FDM-condition" -and will now be further indicatedr From (25) and (28) it follows that~ -~
N
V( W ) = ~ Hm(~))-Tm(~) (31) .
~ ~ 41 - ~ ~
.
10.6.77 ., ~
~ -From (31) and (24) it follows tha~: :
N
V(~J)~= ~ Hm(W )Sm(~)) (32) From (32) and (Z3) it follows that: .
m=l m ~ amkRk(~J) (33)
n X = ~ As ~ Xk(n)~ is a real signal, Xk(~
satisfies the relation:
k( T ) = Xk~~ for ~ ~J 0 < T (17~ :, Suppose that the sampling rate associated ~ -with the desired real FDM signal ~ y(n)~ is equal to - , and that:this sampling rate is an integral .
multiple of T. So assume: T = MT, wherein M ~ N
' and wherein M is an integer. The fundamental inter- :
. val Qy of the frequency spectrum of thi-s FDM-'sig-nal therefor is equal to ~ y = T = Mnx. As .
y(n)~ must be a real signal the frequency spec- ¦
trum Y( W) of this FDM-signal must satisfy the re- :
-h~). Therefor~ this fre-'- ,:
quency spectrum must have in general the form which is diagrammatioally depicted in Figure 20a for N = 4 and M = 5, The channel-signals are each located in a aubband Fk of length T wherein k = 1, 2, 3; N.
Eaçh subband is characterized by the frequency range :
J, + (k-1)~ T~ ~ W < W1 + k.T . In what fo lows here-inafter it will be assumed that 0 h~1 < T . As Y(~J) must represent the frequency spectrum of a FDM-signal which comprises a frequency multiplexr of N baseband signals ~ x(n)~ , the following ex- ,:
pressions must be satisfied: .
~ !
10.6.77 3~ `*
¦ I O ( ) T ~ = Xk(hJo) (18~) and~
[ T - { 1 ~ ~J o + ( k-1) T~
= y~ `[Wl + /~J + (k-l ) ~ l = Xk ~ W O) = Xk ( T o , (18b)l ~ ;
: ~, ~
~: ~ It should be noted that for ~1 = the ~ .:
value of M may be taken equal to N so: M = N,` so . :
~: that the frequency spectrum:Y(~J) has the form de- ~
; .:' . picted in Figure ZOb. .
E(2.2) General implementation of the TDM-FDM-arran~ement :
Figure 21 shows diagrammatically the imple- ~ ;
mentation of a digital TDM-FDM arrangement for con-verting N real digital baseband signals ~xk(n)3 , (k = 1, 2, 3~ ... N; n = .... -2, -1, O, ~ 2~ ...... ), - :
ha~ing associated therewith a sampling period T, into a real digital baseband single-sideband frequency-division multlplex signal ~y(n)~ having associat-ed therewith a sampling period Ty = M. This arrange- ¦
ment comprises an input circuit formed by N input channels 1(1), 1(2), 1(33, .. ~ 1(N). A digital baseband signal { xk(n)~ is applied to each of these input channels. The frequency spectra of these .
. ~ .
P~IN 8731 10.6-77 '~
signals follow from (4) and these spectra are shown ~':
in Figure 19. For generating an FDM-signal having .
the frequency spectrum depicted in Figure 20a for - N = 4 a comp].ex modulator 1(1,1), 1(1,2), .. 1(1,N) ::
is included in each of the input channels and the - ~
input channels having an even number are provided ~ . -with a sideband interchanging modulator 2(1); 2(2); ~ :
.2(3); ... 2(N/2). .The input channels are connected . ,.
to inputs of a transformation device 3. The digital ~ :
signals generated by the input channels and being ~: applied to the inputs of the transformation device .~ -3 will be indicated by frk(n)~ . The relation- -~ .
:-:
~: : ship between the components rk(n) and xk(n) can mnthematically be given by the expression:
rk(n) = xk(n) ei 1 for k is odd k(n) = (-1) xk(n) e~ ~1nT for k is even .~
~, :';, (19) ~ -The frequency spectrum Rk(W ) of the digital signal - :
~rk(n)~ can be derived f~om (14) and (15), as well -.
as from the Figures 14, 15 and 18a and is given by: .
k(~J) = Xk ~ 4 - ~1 ~ (k-l). TW ~ (20) For k is odd it applies that k-1 is even so that k(~J) = Xk(h~ ~ ~J1) for k = odd.
' ~" ~ pl~ 8731 ~ 534 10.6.77 ~ ~
' ` . ~"''' ' ' '' For k is even it applies that ~-1 is odd so that -Rk(W ) = ~ ~ -~J1 - (k-1). T~ ~ for k = even ~ [ 1 T
The spectrum defined by (20) is depicted in Figure 22.
~5 ~ The transformation device 3 produces N di-gital signals ~ sm(n)~ , with m = 1, 2, 3, .,,N, having associated therewith a sampling period T, These digital signals f sm(n)~ are each applied to a signal channel 4(t), 4(2),... 4(N). The proces- :
~l0 sing operation which is performed by the transfor-~ mation device 3 can mathematically be described by ; ~ the expression: ;~
, N
(n) = ~ amk rk( ) (21) ~ ;' ~ ~
~; m 1, 2, 3~ .,.N.
15~ In this expression a k represents a multiplication factor of constant value. This multiplication fac-tor can be a real but also a complex number. If in general it is assumed that amk is a complex number which can be given by: ;~ ~-20~m~ ~ mk i ~ mk (22) then sm(n) also represents a complex number.
. ,~
:~ 38 _ .'' .
:;.
10.6.77 ,~
''~ '~' The general implementation of the transfor- -mer 3 as well as the general implementation of the signal channels 4(m) will be described in chapters E(2.4) and E(2,5). At this moment it is supposed ,:
that the complex signals ~s (n)~ are available in a form which is suitable for further proces-,,: ~..:
sing.
From (213 it is clear that each compo~ ;
~ nent sm(n) i9 formed by a linear combination of com- . ~ ' ponents rk(n). As a result of the linear character `
;
~: of (21) it applies that the frequency spectrum `
m(~J) f~ {Sm(n)~ is given by~
Sm( ~ ) = ~ amk k(W ) (23) The signal channels 4(1), 4(2),;..4(N) each comprise a cascade arrangement of an SRI-ele-ment 5(1), 5(2),...5(N) and a digital filter 6(1), 6(2),.,.6(N) and are connected to inputs of an adder ~' 7, In Figure 21 the components of the OUtpllt signal of the SRI-element 5(m) are indicated by tm(n);
the componentsof the output signal of the digital filter 6(m) are indicated by um(n); and the compo-.
nents of the output signal of the adder 7 are in- -; dicated by v(n). As ~v(n)~ in general represcnts a complex digital signal and as we are only in-` terested in a real digital output signal y(n) ": .
_ 39 , PHN 8731 10.6,77 having the frequency spectrum Y(~ ) which for N = 4 ..
is depicted in Figure ZOa, the components v(n) are ~ .
applied to a selec~-or 8 which only produces the real portion of the complex component v(n) at its output.
In order to find a mathematical expression, for the output signal ~ y(n)~ it should be noted ',:-that the relation between t (n) and sm(n) is given by expression (8). From (10) it follows that the .
, frequency spectrum Tm( W) of ~tm(n)~ is given by:
T (~) = S ( ~ ) : : (24) : T = M Q s = T .. . -M being an integer and M~ N :~ .`.
The digital signal f tm(n)~ is filtered ~: ~ in the filter 6(m). If the transfer function of the :
filter 6(m).is represented by Hm( W ) and the fre-:15 quency ~pectrum of the signal ~ um(n)~ by Um(~
: then it applies that:
.: : .
m(~J) = Hm(~ )Tm( ' (25) m = 1, 2, 3, ...N
The.complex output signal ~ v(n)~ of this TDM-FDM arrangement is now obtained by adding the signals f u~(n)~ together so that:
N :.~
v(n) = ~ um(n) (26) .:
P~IN 8731 ' 10.6.77 and consequently: ~-- N ~ ;
y(n) = Re L ~1 Um(~ (27) : .
The frequency spectrum V( W) of ~ v(n)~
is given by:
.:
N
V( W) = ~ Um(~J) (28) ,~
;
As (27) can be written in the form:
y(n) = Re ~v(n)~ = ~ [v(n)~v (n)~ (~9) wherein v (n') represents the complex conjugate of v(n) and as the frequency spectrum of {v(n)~is equal to V( W ), so that the frequency spectrum of { vX(n)~is equal to V~( T ~ W ) it holds that:
Y :
( j [v(~J) + V ( T ~~ 3)~
E(2.~) The FDM- conditions In order to generate an SSB-FDM-signal having the frequency spectrum Y(W ), which for N = 4 I .:
is depicted in Figure 20a the transfer function H~n( W) --of the filter 6(m) must satisfy a very special con-dition. This condition will be called "FDM-condition" -and will now be further indicatedr From (25) and (28) it follows that~ -~
N
V( W ) = ~ Hm(~))-Tm(~) (31) .
~ ~ 41 - ~ ~
.
10.6.77 ., ~
~ -From (31) and (24) it follows tha~: :
N
V(~J)~= ~ Hm(W )Sm(~)) (32) From (32) and (Z3) it follows that: .
m=l m ~ amkRk(~J) (33)
5~ From (33) and (20) it follows that:
V(w ) = ~ Hm(W ) ~ amkXk [W_ ~1-(k 1) T
. ' (34) (34)can also be written in the form:
V(~J) = ~ Xk [W -iJ~ (k 1) T 3 ~ ~mk m( , 80 thats N
V~(~ _ W) = ~ X ~ - ~_ Wl-(k-1)T ~ .
{ amkHm(~Qy- ~) ( ) ~' !
As ~ = M ~ = M 2T ~ from (30), (35) and (36) it follows that:
'';
~ - 42 10.6.77 ';"'""
Y( W ) = ~ ~ Xk [W~ (k-1) T ] . ~ amkHm(~J,) ~ I
. . ~ ~':"' ~( Q ~) ¦
(37) As can be easily verified Yt~3 as defined in (37) ~ ~-indeed satisfies the condition (6) relating to a real output signal f y(n)~ . From (37) it follo~s indeed that: : 1:
f' `'`
Y(ny W) = ~(~) Therefore it is sufficient to investigate which . . :~
~10 : oondition the transfer function Hm(~J) must satis- :
. fy ln the various subbands Fi of bandwidth T~
(i = 1, 2, 3, .... N) in order to obtain the desired :
frequency spectrum Y(W ). Suppose ; J = ~0 + ~1 + (i-1) T
~ with ~ ~ < T- :~
and i = 1, 2, 3, ... N.
- Then (37) passes into:
+ l + (i-l) T ~ = ~ ~ Xk ~W 0 + (i-k) T ~
mkHm ~o + ~J 1 ~ (1-1) T + ~ --. ~., .
. - 43 . , . . ., . .. , , , , ~
PHN 873l ~$~3~ 10.6.77 -.
', ' k [~O + 2 ~ ~ + (i--k) ~ ] . ~ :
mkHm rM- T ~ { ~0 + ~1 + (i-1) T } ] (38) ~
In accordance with (18a) the expression: : ~.
. Y [ 4 + ~J1 + (i-1) T ] = Xl ( W0) (39) ~- ;
~"
must now be satisfied. . .
Two situations can now be distinguished, namely the :
; situations: :
~ ~ 1. W1 = ' ~`
. . .
~' ~ Z. W~ ;~ O ` '`' ' ~ 1. The situatlon ~J1 = means that the ~.
TDM-FDM arrangement shown in Figure 21 is arrang-ed for generating the SSB-FDM signal having the frequency spectrum which is depicted in Figure 20b.
. : As has already been remarked ln chapter E(2.1) the ihcrease ~actor M of the SRI-elements can now be chosen to be equal to N so: M=N. With these data it follows from (38) that: ¦~
~; r k=1 ~ ~amkHm [W0 + (1-1) T ]
+ amkHm r ~ ~ T ~ ~W0 + (i-1) T~ } l} (4) -: - 44 - :.
, lo.6.77 34 ::
`"~
As the relation (see (39)) Y [ W 0 + (i-1) T ~ - Xi(40) must now be satisfied (see(39)); we obtain from (40) the FDM-condition for Wl~= 0:.
.... .
mk~m [~0 ~ (i-1) T- J + a kH~ ~;
rM 2~ ~ + (i-1) T ~ ki 2. The situation ~J1 ~ means that the TDM-FDM arrangement shown in Figure 21 is arranged for generating the SBB-FDM signal having the frequency spectrum whlch is depicted in Figure 20a. In this : case (38) can only satisfy (39) if the FDM-condi-tion for W ~ o satisfies which reads as follows:
Hm ~M- T . { 0 + 1 ( ) T ~]
(42) ~ ~1 amk Hm ~ 0 + W 1 + (i-1) T ~
In (41) and (42) ~ ki represents the Kronecker sym-bol with is defined as follows: : ` :
ki = for k ~ i =1 for k = i '~
10.6.77 -S~
i Although the frequency spectrum of the FDM
signal within the frequency range O ~ W ~ M ~ is taken into account only to deri~e the expressions, (41) and (42), these expressions are ~alid for any arbitrary frequency range ~ M. ~- ~ W ~ (0~ ~ 1) M. T
This follows from the fact that both Y(~i) and Hm( W) are periodic functions with period M. T- so that ex-pression (5) applies. Consequently, the FDM-condi-tion (41) mi~ht also be written in the equivalent form~
~, ~1 { mkHm rlXN--T + ~Jo + ~1 + (i-l) ~] I
. . . :~ -2~ ~ ~
+ amkHm [(~ +1)-N- ~ ~ { o+~l~(i l)T }]~
= ~ (44) ki wherein ~ represents an integer, The multiplication factors amk may be considered as the elements of a NxN-matrix A(N).
Herein N indicates the order of the matrix. This matrix then has the form:
11 12 13....a (N) ~ ] ~ 21 22 a23~ a aN1 aN2 aN3........ aNN -' ~:
:` ~ ~:`
~ 1o.6.77 `~' and will be called transfOrmation-matrix. ~:
~ o < T ~ the function Hm r W + ~J
+ (i-1) T ~ which is part of expression (38) ,~
describes the transfer function Hm(~J) over the fre-quency range ~J1 + (i-1~. T ~J ~ ~J~ These functions may be considered to be elements of a NxN-matrix H( W ). This matrix has the form:
.. ~ ;''~ ' ~ H( 0) ~ :
` '.'.~
.
1( o~ 1) 1( o~1+T)~ H1~o'~ r~\ ~ -~
2 t O t ) H2(C~O+~1 +~ H ~ cc~ * 6~1 * (~ ' . . . . (46) . HN(c-~ ~ 71 ) N( o~ 1 +T) ~ INr ~'70*
.~
.. _ __.... . .. . ..... _ ...... , . . . _ ..... : ;:~
; and will be called transfer-matrix. .
In a similar manner m ~ ~T {~J ~ ~1 + (i-1) T-}]with i = 1, 2, 3, ..., N
describes the transfer function H ( W ) over the fre-quency range (2M-i) T ~ W 1 ~ ~J< (2M - i+1) T ~ ~1 ~-I ,~
- l~7 -1~14S34 ': ';~ , These last-mentioned functions may be considered to be ele-ments of an NxN-matrix _(M.2T - O). This matrix has the form:
_(2MT~ - ~0) = :
5~ Hl[2MT~ -wl-~o] H1[(2M-l)T~r-Wl-Wo]~Hl[(2M~N+l)T-H [2M~ -~1-~ ] H2[(2M-l)T- -wl ~o]..H2[( T 1 o ~47) ~ ' J ''"','.
HN[2MT~r-~l-~o] HN[(2M-1)2~-~1-wo].HN[(2M-N+l)T -~l-~o]
If in the expressions (46) and (47) it is supposed that~l = O then the FDM-condition (41) for ~Jl = can be written in the form:
A(N)T H(~ ) + (A(N) ) H (2M T ~ ~0) = 2IN (48) With (46) and (47) the FDM-condition (42) for ~1 ~ can now be written in the form:
H(2M T ~ ~o) =
A(N)T . ~I(~o) = 2IN
In the expressions (48) and (49):
A(N) represents the transposed matrix of A(N) ;
A(N)X represents the complex conjugate matrix of A(N) ;
;~ .
pl~ 873l 10.6.77 45~ - .;
, IN represents the NxN-identity matrix.
When deducing the two FDM-conditions (41) ~ `~
and (42) it was assumed that the ~D~1-signal satis-fies (39). If, however, a given amplitude and phase distortion is allowed ~or the channel signal in the SSB-FDM signal relative to the original baseband signal then this can be expressed by writing (39) ~.
in the form~
Y ~ + ~1 + ( i- 1 ) T--` ~ = Xi ( ~J o ) ~i ( ~ :
:"- , Herein ~ Jo) represents a function of W . This am- : .
plitude and phase distortion can now also be expressed in the FDM-condition, namely by replacing in expres-~ ~
sions (41) and (42) 6 ki by ~ki- ~ o) Y ~:
- placing in expressions (48) and (49) IN by diag [ ~ i( ~ o))~ Herein diag ~ ~ i(h~o)~ ls defined as follows:
.. . . , .. ,. . . ., . . .............. _ _ _ __~ __ _, ._ __ _._ . ~ .
. ' . ''' ~(~0) ' O.... 0 \' ' '' .
/ ~2 ( ¢~o ) - - - - O \ ' .
g~i(~o)~=¦ ~ (~ ) (51) -.
- I :
O O . O ----~
. . ,, ,, . . . I
When deducing the expressions (41) and (42) :
it was assumed that the output signal y(n) of the ;~
~, _ 49 _ ~ -PHN 8,31 10.6.77 ' TDM-F~M arrangement is formed by the signal Re ~ v(n)~ . It will be clear that also the signal -Im [ v(n)] could be taken as the output signal of this TDM-FDM arrangement. The two FDM-conditions (48) and (49) apply also for this choice of the output signal.
E(2,4) The transformation device The transformation device 3 of Figure 21 is arranged for performing the operations defined in (21). This transformation device is based on the matrix A( ) defined in (45). As the matrix A( ) is of the N-order the tranfiformation-device will be indicated as N-point transformer. In agree~ent herewith the TDM-FDM-arrangement w~lich is provided with a N-point transformer will be indicated as N-point TDM-FDM arrangement.
When the multiplication factors amk in expression (21) re given by (22) then (21) can be written in the form.
N N
sm(n) = ~ CXmk'rk(n) ~ mk.rk(n) (52) m = 1, 2~ 3...N.
Thus the component sm(n) has a real portion and an imaginary portion. If the real portion of sm(n) is represented by Re { sm(n)~ and the imaginary ~ -~
portion by Im r sm(n)] then it holds that:
, 114~34 1o.G.77 ~
. .
s (n) = Re ~s (n)~ + j Im ~sm(n)]
: , .
wherein:
[sm(n)] = ~ C~ rk(n) ~' [ m(n)~ mk.rk(n) To be able to process a complex signal the real portion and the imaginary portion of this sig-nal must be available separately. Figure 23 now shows a transformation-device which is arranged for : - ~
~ generating the components, defined in (53) in case ,. . .
N=2 and its input signals are real. This two-point , ~' transformer comprises two inputs 1(1) and 1(2).
Connected to each of these inputs there are four multipliers 9(~ ), 10(~ = 1,2~3~4) which mul-- tiply the input components rl(n) and r2(n) applied theret-o by the multiplication factors ~ mk and mk-The outputs of the multipliers are con-nected in the way shown in the Figure to inputs of adders 11(~ ) wlth ~ = 1,2,3,4, the outputs where- ;
of form the outputs of the transformation-device.
The adder 11 ( 1 ) produces the real part Re [ sl(n)] of 51 (n) and the adder 11(2) the imaginary part Im ~ s1(n)~ of s1(n).
.
~.
0.~.77 , ~
' Following the above teachings it is simple `
to deduce the implementation of a transformation-de-vice in case N > 2. The implementation thus obtained is usually called the ~direct implementation" (of expression (52)). .
The transformation-device of Figure 23 i9 actually a special embodiment of a general transfor- ~ 3 mation-device which is arranged for converting com-plex input components into complex output components and which is based on a transformation matrix whose .
elements are complex numbers. If it is assumed that the input components rk(n) are given by ` -rk(n) = Re ~ rk(n)~ + ; Im r rk(n)]
. . '.' - "' and that the multiplication factors amk are again glven by (22) then (21) can be written as in the form. ' ' .:
s~(n).= ~ fCmk-Re rrh(n)~ ~ ~ mk.Im [rk(n)~}
~1 fc~mk Im ~rk(n)] ~ ~ mk-Re rrk(n)]~ (54) sm(n) = Re rsm(n)] + J Im ~Sm(~)]
In F~gure 24 this general transformation-device is shown for the case N = 2. Thls general two-point _ 52 -:. .
~HN 8731 ~ 10.6.77 ` , transformer is provided with four inputs 1(1,1), ~-1(1,2), 1(2,1) and 1(2,2) which are each connected ~ -to inputs of four multipliers 12(~ ), 13(y ), ~ ;
14(~ )-and 15( ~ ) with ~ = 1,2,3,4 which multiply -the input components applied thereto by the multi- ~
plication factorS ~ mk' ~ mk~' ~ mk P ~v of these multipliers are furthermore connected to ~ ~-inputs of adders 16( ~) with ~ = 1,2,3,4, whose outputs again constitute the outputs of the trans-formation-device.
Following the above teaching it is again simple to deduce the lmplementation of the general -~
tran~formation-device in case N ~ 2.
.: -From the above it follows that the number of complex multiplications which must be performed in the transformation-device in order to calculate one output component y(n) of the TDM-FDM arrange-ment~ is equal to N2. For the evaluation of ex-. .
pression (52? this means 2N`2 ~eal multiplications and for the evaluation of expression (54) this ;
means 4N real multiplications. The complexity of ~ ::
the TDM-FDM arrangement is now inter alia deter-- - :
~ mined by the value of N.
,:
In a manner reminiscent of the discrete fourier transform (DFT), (see reference 13) a transformation-device whose operation is fully described by a matrix can be implemented in given ' $3 10.6.77 '~
circumstances in such a way that the number of mul-tiplications to be performed is drastically reduc- -ed. A transformation-device implernented in such a manner will be called "fast transformation-device".
The fast transformation-device for calculating the ' discrete fourier transform is, for example, kno~
as ,"Fast Fourier TransfoFmer (FFT).
Such a fast transformation-device may, for ~ , example, be implement,ed using two-point transformers " ~, -' 10 which are based on a matrix Aiz) and which are each implemented in the manner as, for example, sho~m in ,'~
Figure 23 or 24.'The two-point transformer of Figure ~, 23 and of Figure 24 respectively will hereinafter be indicated by the symbol sho~m in Figure 25 and Figure ,, 26 respectively. In these symbols the associated ~' transformation matrices are indicated by Ai2z) and this matrix is given by: '~
" ' Aiz ( ) (55) ;20 ' aiz,21 aiz,22 Figure 27 shows for completeness' sake ~ "~
an eight-point fast transformation-device which is suitable for use in the arrangement of Figure 21.
The implementation of this transformation-device is based on two-point transformers of the type shol~n in Figure 24 and Figure 26 respectively. These , ,54 . .
:;"'`:` . ~
PHN 8~31 10.6.77 ;:, -transformersare interconnected in a manner ~hown ln the -~
figure.The transformers 3(1), 3(2), 3(3) and 3(4) are -; . based on the matrices A(1), A(1), A31) and A(1) res- ~ r pectively. ~ach of the transformers 3(5,1) and 3(5,2) i8 based on the matrix A522). Last-mentioned transfor-mers may be considered as a transformer 3(5) which .: .
: ~ is based on a matrix which will be sy~bolically . indicated by 2xA(22). Each of the transformer .
: 3(6,1) and 3(6,2) is based on the matrix A(2) so that also these transformers may be considered as ~ ~.
a transformer 3(6) which is based on a matrix .;
~: which will be indicated, ,~gain symbolically, by 2xA(2). Each of the transformers 3(7,1),3(7,2), ; 3(7,3) and 3(7,4) is based on the matrix A(23) so ;.
that these transformers together may be consider- :
ed as a transformer 3(7) which is based on a matrix :
which will be indicated symbolically by 4xA(32). It :: .
should be noted that in the symbol A(2) .
iz the index,~:
z lndicates the column in which the relevant trans- ~. .
former can be found (cf. Figure 27).
It is noted that the implementation of a , ~
fast transformation device can also be based on, for example, four-point transformers, each of : ~ which is based on a matrix Ai4). For its equi-valent for the DFT reference is again made to : : .
reference .13.
' Pl-~ 8731 .~
1~4~ 10.6.77 . '', '"
E(2,5~ The si~nal channel E(2 5.1) General implementation.
In chapter E(2,3) it was assumed that the -~ :
transformation device 3 produces complex signal com-ponents sm(n) having a real part Re [ sm(n)~ and an imaginary part Im [ sm(n)~ so that:
Sm(n) = Re [sm(n)~ + j Im ~sm(n)~
These signal components are applied to the SRI-elements whose complex output signal components ~ .
tm(n) may be represented by `-~
tm(n) = Re [tm(n)~ + j Im [ tm(n)]
In the digital filter 6(m) these components tm(n) are convolved with the impulse response h (n), :
so that:
um(n) = tm(n)x hm(n) (56) wherein um(n) = Re Lum(n)] + j Im [ um(n)] (57) In general H~ (~J) will be unequal to H~(M. T ~ hJ )- This means that the impulse response h (n) is complex which will be repre-. 20 sented as follows:
: hm(n) = hmp(n) + i hmq( ) (58) Hereln hmp(n) and hmq(n) represent real PHN 8731 ~ ~
1~14~34 10.6,77 :
,:
impulse responses.
For the ~urther analysis of the transfer : -function H (W ) transfer functions Hmp(~) and - :
Hmq(~J) are introduced. Herein it is assumed that h~ (n) represents the impulse response of a digi- ~ ~
tal filter having the trans~er function Hm (~J) : :
and that hmq(n) represents the pulse response of -a digital filter having the transfer function H q(~ so that: :
.;
Hm( W) = Hmp( W? + j Hmq( and:
p(~ Hm(~J) + H~(2 ~ _~J)] (60) Hmq( W ) = aJ ~ Hm( ~ ) Hm( M~
The transfer matrices (46) and (47) can now be written respectively as:
H(~Jo) = Hp(~Jo) + i _q( o) (61) ~ .
H(2MT - ~0) = Hp(~o) + j _q( 0) wherein in a corresponding manner as in (46). :.~
', ~*
:.: ~ -; . ;' _ 57 11~4.5~4 ~ lP(l~)o ~)1) Hlp(~)o+~l+ T).Hlp[~o+~+(N~l)~]~
Hp (~0)= 2p (~Jo 1) H2P (~)o+~l+ T) H2p [~o+~Jl+ (N-l)Z] (62) -\ +~ ~' /
HNp(b.~o 1) HNp(h)o+ l+ T)...HNp[~)o+l~Jl+(N~l)~]
',". ~.~, -.
H (~ +~1) Hlq(~o+~l+ ~) Hlq[ 1 T
¦ H2q (~o+~l) H2q (~o+~l+ ~Z) H2 [L~ +~Jl+ (N--1)~] (63) ~ ~ +~+ N-l)~]/ ~`''':
H (~ +~1) HNq(~o+ l+ ~)---HNq[~o 1 ( T
The transfer matrices Hp (2M -~J - ~0) and Hq (2M T ~ ~0) can be defined in accordance with (47). `
'..
; ~;,.:
- ' .
`~ . Pll~ 87~l ~
~3~ 10.6.77 ~
The implementation of a digital filter with ..
complex impulse response now follows directly from (56), ~ :
- (57) and (58). From the~eexpressions it follows that:
Re Lum(n)~ = Re ~tm(n)] ~ hmp(n) - Im [tm(n)] ~ hmq(n 5 . (64) ::.
Im rUm(n)~ = Re [tm(n)] ~ hmq(N) + Im ~tm(n)] ~ h (n The complete implementation of the signal :~
channel 4(m) is shown in Figure 28.
In chapter E(2~4) it is already described that the transformation-device 3 produces the real :~;
part Re [sm(n)] and the imaginary part Im L sm(n)~ ~l`
: of sm(n) on separate outputs. For processing this complex signal {Sm(n)} in the signal channel 4(m) : .
this signal channel comprises two auxiliary channels 4(m,1) and 4(m,2) to which the signals Re [~m(n)]
and Im tsm(n)] are applied respectively (see Fi-.gure 28). Each of these auxiliary channels comprises ~ :*
an SRI-element 5(m,1) and 5(m,2) respectively at the -:
outputs of which the real part Re ~t~n(Il)~ and the imaginary part Im [tm(n)] of the complex signal ~ -[tm(n)~ are produced respectivaly, Said last outputs are furthermore connected to inputs of the digital filter 6(m). In general this filter com-prises four reali~able digital filters 6(m,1), .
V(w ) = ~ Hm(W ) ~ amkXk [W_ ~1-(k 1) T
. ' (34) (34)can also be written in the form:
V(~J) = ~ Xk [W -iJ~ (k 1) T 3 ~ ~mk m( , 80 thats N
V~(~ _ W) = ~ X ~ - ~_ Wl-(k-1)T ~ .
{ amkHm(~Qy- ~) ( ) ~' !
As ~ = M ~ = M 2T ~ from (30), (35) and (36) it follows that:
'';
~ - 42 10.6.77 ';"'""
Y( W ) = ~ ~ Xk [W~ (k-1) T ] . ~ amkHm(~J,) ~ I
. . ~ ~':"' ~( Q ~) ¦
(37) As can be easily verified Yt~3 as defined in (37) ~ ~-indeed satisfies the condition (6) relating to a real output signal f y(n)~ . From (37) it follo~s indeed that: : 1:
f' `'`
Y(ny W) = ~(~) Therefore it is sufficient to investigate which . . :~
~10 : oondition the transfer function Hm(~J) must satis- :
. fy ln the various subbands Fi of bandwidth T~
(i = 1, 2, 3, .... N) in order to obtain the desired :
frequency spectrum Y(W ). Suppose ; J = ~0 + ~1 + (i-1) T
~ with ~ ~ < T- :~
and i = 1, 2, 3, ... N.
- Then (37) passes into:
+ l + (i-l) T ~ = ~ ~ Xk ~W 0 + (i-k) T ~
mkHm ~o + ~J 1 ~ (1-1) T + ~ --. ~., .
. - 43 . , . . ., . .. , , , , ~
PHN 873l ~$~3~ 10.6.77 -.
', ' k [~O + 2 ~ ~ + (i--k) ~ ] . ~ :
mkHm rM- T ~ { ~0 + ~1 + (i-1) T } ] (38) ~
In accordance with (18a) the expression: : ~.
. Y [ 4 + ~J1 + (i-1) T ] = Xl ( W0) (39) ~- ;
~"
must now be satisfied. . .
Two situations can now be distinguished, namely the :
; situations: :
~ ~ 1. W1 = ' ~`
. . .
~' ~ Z. W~ ;~ O ` '`' ' ~ 1. The situatlon ~J1 = means that the ~.
TDM-FDM arrangement shown in Figure 21 is arrang-ed for generating the SSB-FDM signal having the frequency spectrum which is depicted in Figure 20b.
. : As has already been remarked ln chapter E(2.1) the ihcrease ~actor M of the SRI-elements can now be chosen to be equal to N so: M=N. With these data it follows from (38) that: ¦~
~; r k=1 ~ ~amkHm [W0 + (1-1) T ]
+ amkHm r ~ ~ T ~ ~W0 + (i-1) T~ } l} (4) -: - 44 - :.
, lo.6.77 34 ::
`"~
As the relation (see (39)) Y [ W 0 + (i-1) T ~ - Xi(40) must now be satisfied (see(39)); we obtain from (40) the FDM-condition for Wl~= 0:.
.... .
mk~m [~0 ~ (i-1) T- J + a kH~ ~;
rM 2~ ~ + (i-1) T ~ ki 2. The situation ~J1 ~ means that the TDM-FDM arrangement shown in Figure 21 is arranged for generating the SBB-FDM signal having the frequency spectrum whlch is depicted in Figure 20a. In this : case (38) can only satisfy (39) if the FDM-condi-tion for W ~ o satisfies which reads as follows:
Hm ~M- T . { 0 + 1 ( ) T ~]
(42) ~ ~1 amk Hm ~ 0 + W 1 + (i-1) T ~
In (41) and (42) ~ ki represents the Kronecker sym-bol with is defined as follows: : ` :
ki = for k ~ i =1 for k = i '~
10.6.77 -S~
i Although the frequency spectrum of the FDM
signal within the frequency range O ~ W ~ M ~ is taken into account only to deri~e the expressions, (41) and (42), these expressions are ~alid for any arbitrary frequency range ~ M. ~- ~ W ~ (0~ ~ 1) M. T
This follows from the fact that both Y(~i) and Hm( W) are periodic functions with period M. T- so that ex-pression (5) applies. Consequently, the FDM-condi-tion (41) mi~ht also be written in the equivalent form~
~, ~1 { mkHm rlXN--T + ~Jo + ~1 + (i-l) ~] I
. . . :~ -2~ ~ ~
+ amkHm [(~ +1)-N- ~ ~ { o+~l~(i l)T }]~
= ~ (44) ki wherein ~ represents an integer, The multiplication factors amk may be considered as the elements of a NxN-matrix A(N).
Herein N indicates the order of the matrix. This matrix then has the form:
11 12 13....a (N) ~ ] ~ 21 22 a23~ a aN1 aN2 aN3........ aNN -' ~:
:` ~ ~:`
~ 1o.6.77 `~' and will be called transfOrmation-matrix. ~:
~ o < T ~ the function Hm r W + ~J
+ (i-1) T ~ which is part of expression (38) ,~
describes the transfer function Hm(~J) over the fre-quency range ~J1 + (i-1~. T ~J ~ ~J~ These functions may be considered to be elements of a NxN-matrix H( W ). This matrix has the form:
.. ~ ;''~ ' ~ H( 0) ~ :
` '.'.~
.
1( o~ 1) 1( o~1+T)~ H1~o'~ r~\ ~ -~
2 t O t ) H2(C~O+~1 +~ H ~ cc~ * 6~1 * (~ ' . . . . (46) . HN(c-~ ~ 71 ) N( o~ 1 +T) ~ INr ~'70*
.~
.. _ __.... . .. . ..... _ ...... , . . . _ ..... : ;:~
; and will be called transfer-matrix. .
In a similar manner m ~ ~T {~J ~ ~1 + (i-1) T-}]with i = 1, 2, 3, ..., N
describes the transfer function H ( W ) over the fre-quency range (2M-i) T ~ W 1 ~ ~J< (2M - i+1) T ~ ~1 ~-I ,~
- l~7 -1~14S34 ': ';~ , These last-mentioned functions may be considered to be ele-ments of an NxN-matrix _(M.2T - O). This matrix has the form:
_(2MT~ - ~0) = :
5~ Hl[2MT~ -wl-~o] H1[(2M-l)T~r-Wl-Wo]~Hl[(2M~N+l)T-H [2M~ -~1-~ ] H2[(2M-l)T- -wl ~o]..H2[( T 1 o ~47) ~ ' J ''"','.
HN[2MT~r-~l-~o] HN[(2M-1)2~-~1-wo].HN[(2M-N+l)T -~l-~o]
If in the expressions (46) and (47) it is supposed that~l = O then the FDM-condition (41) for ~Jl = can be written in the form:
A(N)T H(~ ) + (A(N) ) H (2M T ~ ~0) = 2IN (48) With (46) and (47) the FDM-condition (42) for ~1 ~ can now be written in the form:
H(2M T ~ ~o) =
A(N)T . ~I(~o) = 2IN
In the expressions (48) and (49):
A(N) represents the transposed matrix of A(N) ;
A(N)X represents the complex conjugate matrix of A(N) ;
;~ .
pl~ 873l 10.6.77 45~ - .;
, IN represents the NxN-identity matrix.
When deducing the two FDM-conditions (41) ~ `~
and (42) it was assumed that the ~D~1-signal satis-fies (39). If, however, a given amplitude and phase distortion is allowed ~or the channel signal in the SSB-FDM signal relative to the original baseband signal then this can be expressed by writing (39) ~.
in the form~
Y ~ + ~1 + ( i- 1 ) T--` ~ = Xi ( ~J o ) ~i ( ~ :
:"- , Herein ~ Jo) represents a function of W . This am- : .
plitude and phase distortion can now also be expressed in the FDM-condition, namely by replacing in expres-~ ~
sions (41) and (42) 6 ki by ~ki- ~ o) Y ~:
- placing in expressions (48) and (49) IN by diag [ ~ i( ~ o))~ Herein diag ~ ~ i(h~o)~ ls defined as follows:
.. . . , .. ,. . . ., . . .............. _ _ _ __~ __ _, ._ __ _._ . ~ .
. ' . ''' ~(~0) ' O.... 0 \' ' '' .
/ ~2 ( ¢~o ) - - - - O \ ' .
g~i(~o)~=¦ ~ (~ ) (51) -.
- I :
O O . O ----~
. . ,, ,, . . . I
When deducing the expressions (41) and (42) :
it was assumed that the output signal y(n) of the ;~
~, _ 49 _ ~ -PHN 8,31 10.6.77 ' TDM-F~M arrangement is formed by the signal Re ~ v(n)~ . It will be clear that also the signal -Im [ v(n)] could be taken as the output signal of this TDM-FDM arrangement. The two FDM-conditions (48) and (49) apply also for this choice of the output signal.
E(2,4) The transformation device The transformation device 3 of Figure 21 is arranged for performing the operations defined in (21). This transformation device is based on the matrix A( ) defined in (45). As the matrix A( ) is of the N-order the tranfiformation-device will be indicated as N-point transformer. In agree~ent herewith the TDM-FDM-arrangement w~lich is provided with a N-point transformer will be indicated as N-point TDM-FDM arrangement.
When the multiplication factors amk in expression (21) re given by (22) then (21) can be written in the form.
N N
sm(n) = ~ CXmk'rk(n) ~ mk.rk(n) (52) m = 1, 2~ 3...N.
Thus the component sm(n) has a real portion and an imaginary portion. If the real portion of sm(n) is represented by Re { sm(n)~ and the imaginary ~ -~
portion by Im r sm(n)] then it holds that:
, 114~34 1o.G.77 ~
. .
s (n) = Re ~s (n)~ + j Im ~sm(n)]
: , .
wherein:
[sm(n)] = ~ C~ rk(n) ~' [ m(n)~ mk.rk(n) To be able to process a complex signal the real portion and the imaginary portion of this sig-nal must be available separately. Figure 23 now shows a transformation-device which is arranged for : - ~
~ generating the components, defined in (53) in case ,. . .
N=2 and its input signals are real. This two-point , ~' transformer comprises two inputs 1(1) and 1(2).
Connected to each of these inputs there are four multipliers 9(~ ), 10(~ = 1,2~3~4) which mul-- tiply the input components rl(n) and r2(n) applied theret-o by the multiplication factors ~ mk and mk-The outputs of the multipliers are con-nected in the way shown in the Figure to inputs of adders 11(~ ) wlth ~ = 1,2,3,4, the outputs where- ;
of form the outputs of the transformation-device.
The adder 11 ( 1 ) produces the real part Re [ sl(n)] of 51 (n) and the adder 11(2) the imaginary part Im ~ s1(n)~ of s1(n).
.
~.
0.~.77 , ~
' Following the above teachings it is simple `
to deduce the implementation of a transformation-de-vice in case N > 2. The implementation thus obtained is usually called the ~direct implementation" (of expression (52)). .
The transformation-device of Figure 23 i9 actually a special embodiment of a general transfor- ~ 3 mation-device which is arranged for converting com-plex input components into complex output components and which is based on a transformation matrix whose .
elements are complex numbers. If it is assumed that the input components rk(n) are given by ` -rk(n) = Re ~ rk(n)~ + ; Im r rk(n)]
. . '.' - "' and that the multiplication factors amk are again glven by (22) then (21) can be written as in the form. ' ' .:
s~(n).= ~ fCmk-Re rrh(n)~ ~ ~ mk.Im [rk(n)~}
~1 fc~mk Im ~rk(n)] ~ ~ mk-Re rrk(n)]~ (54) sm(n) = Re rsm(n)] + J Im ~Sm(~)]
In F~gure 24 this general transformation-device is shown for the case N = 2. Thls general two-point _ 52 -:. .
~HN 8731 ~ 10.6.77 ` , transformer is provided with four inputs 1(1,1), ~-1(1,2), 1(2,1) and 1(2,2) which are each connected ~ -to inputs of four multipliers 12(~ ), 13(y ), ~ ;
14(~ )-and 15( ~ ) with ~ = 1,2,3,4 which multiply -the input components applied thereto by the multi- ~
plication factorS ~ mk' ~ mk~' ~ mk P ~v of these multipliers are furthermore connected to ~ ~-inputs of adders 16( ~) with ~ = 1,2,3,4, whose outputs again constitute the outputs of the trans-formation-device.
Following the above teaching it is again simple to deduce the lmplementation of the general -~
tran~formation-device in case N ~ 2.
.: -From the above it follows that the number of complex multiplications which must be performed in the transformation-device in order to calculate one output component y(n) of the TDM-FDM arrange-ment~ is equal to N2. For the evaluation of ex-. .
pression (52? this means 2N`2 ~eal multiplications and for the evaluation of expression (54) this ;
means 4N real multiplications. The complexity of ~ ::
the TDM-FDM arrangement is now inter alia deter-- - :
~ mined by the value of N.
,:
In a manner reminiscent of the discrete fourier transform (DFT), (see reference 13) a transformation-device whose operation is fully described by a matrix can be implemented in given ' $3 10.6.77 '~
circumstances in such a way that the number of mul-tiplications to be performed is drastically reduc- -ed. A transformation-device implernented in such a manner will be called "fast transformation-device".
The fast transformation-device for calculating the ' discrete fourier transform is, for example, kno~
as ,"Fast Fourier TransfoFmer (FFT).
Such a fast transformation-device may, for ~ , example, be implement,ed using two-point transformers " ~, -' 10 which are based on a matrix Aiz) and which are each implemented in the manner as, for example, sho~m in ,'~
Figure 23 or 24.'The two-point transformer of Figure ~, 23 and of Figure 24 respectively will hereinafter be indicated by the symbol sho~m in Figure 25 and Figure ,, 26 respectively. In these symbols the associated ~' transformation matrices are indicated by Ai2z) and this matrix is given by: '~
" ' Aiz ( ) (55) ;20 ' aiz,21 aiz,22 Figure 27 shows for completeness' sake ~ "~
an eight-point fast transformation-device which is suitable for use in the arrangement of Figure 21.
The implementation of this transformation-device is based on two-point transformers of the type shol~n in Figure 24 and Figure 26 respectively. These , ,54 . .
:;"'`:` . ~
PHN 8~31 10.6.77 ;:, -transformersare interconnected in a manner ~hown ln the -~
figure.The transformers 3(1), 3(2), 3(3) and 3(4) are -; . based on the matrices A(1), A(1), A31) and A(1) res- ~ r pectively. ~ach of the transformers 3(5,1) and 3(5,2) i8 based on the matrix A522). Last-mentioned transfor-mers may be considered as a transformer 3(5) which .: .
: ~ is based on a matrix which will be sy~bolically . indicated by 2xA(22). Each of the transformer .
: 3(6,1) and 3(6,2) is based on the matrix A(2) so that also these transformers may be considered as ~ ~.
a transformer 3(6) which is based on a matrix .;
~: which will be indicated, ,~gain symbolically, by 2xA(2). Each of the transformers 3(7,1),3(7,2), ; 3(7,3) and 3(7,4) is based on the matrix A(23) so ;.
that these transformers together may be consider- :
ed as a transformer 3(7) which is based on a matrix :
which will be indicated symbolically by 4xA(32). It :: .
should be noted that in the symbol A(2) .
iz the index,~:
z lndicates the column in which the relevant trans- ~. .
former can be found (cf. Figure 27).
It is noted that the implementation of a , ~
fast transformation device can also be based on, for example, four-point transformers, each of : ~ which is based on a matrix Ai4). For its equi-valent for the DFT reference is again made to : : .
reference .13.
' Pl-~ 8731 .~
1~4~ 10.6.77 . '', '"
E(2,5~ The si~nal channel E(2 5.1) General implementation.
In chapter E(2,3) it was assumed that the -~ :
transformation device 3 produces complex signal com-ponents sm(n) having a real part Re [ sm(n)~ and an imaginary part Im [ sm(n)~ so that:
Sm(n) = Re [sm(n)~ + j Im ~sm(n)~
These signal components are applied to the SRI-elements whose complex output signal components ~ .
tm(n) may be represented by `-~
tm(n) = Re [tm(n)~ + j Im [ tm(n)]
In the digital filter 6(m) these components tm(n) are convolved with the impulse response h (n), :
so that:
um(n) = tm(n)x hm(n) (56) wherein um(n) = Re Lum(n)] + j Im [ um(n)] (57) In general H~ (~J) will be unequal to H~(M. T ~ hJ )- This means that the impulse response h (n) is complex which will be repre-. 20 sented as follows:
: hm(n) = hmp(n) + i hmq( ) (58) Hereln hmp(n) and hmq(n) represent real PHN 8731 ~ ~
1~14~34 10.6,77 :
,:
impulse responses.
For the ~urther analysis of the transfer : -function H (W ) transfer functions Hmp(~) and - :
Hmq(~J) are introduced. Herein it is assumed that h~ (n) represents the impulse response of a digi- ~ ~
tal filter having the trans~er function Hm (~J) : :
and that hmq(n) represents the pulse response of -a digital filter having the transfer function H q(~ so that: :
.;
Hm( W) = Hmp( W? + j Hmq( and:
p(~ Hm(~J) + H~(2 ~ _~J)] (60) Hmq( W ) = aJ ~ Hm( ~ ) Hm( M~
The transfer matrices (46) and (47) can now be written respectively as:
H(~Jo) = Hp(~Jo) + i _q( o) (61) ~ .
H(2MT - ~0) = Hp(~o) + j _q( 0) wherein in a corresponding manner as in (46). :.~
', ~*
:.: ~ -; . ;' _ 57 11~4.5~4 ~ lP(l~)o ~)1) Hlp(~)o+~l+ T).Hlp[~o+~+(N~l)~]~
Hp (~0)= 2p (~Jo 1) H2P (~)o+~l+ T) H2p [~o+~Jl+ (N-l)Z] (62) -\ +~ ~' /
HNp(b.~o 1) HNp(h)o+ l+ T)...HNp[~)o+l~Jl+(N~l)~]
',". ~.~, -.
H (~ +~1) Hlq(~o+~l+ ~) Hlq[ 1 T
¦ H2q (~o+~l) H2q (~o+~l+ ~Z) H2 [L~ +~Jl+ (N--1)~] (63) ~ ~ +~+ N-l)~]/ ~`''':
H (~ +~1) HNq(~o+ l+ ~)---HNq[~o 1 ( T
The transfer matrices Hp (2M -~J - ~0) and Hq (2M T ~ ~0) can be defined in accordance with (47). `
'..
; ~;,.:
- ' .
`~ . Pll~ 87~l ~
~3~ 10.6.77 ~
The implementation of a digital filter with ..
complex impulse response now follows directly from (56), ~ :
- (57) and (58). From the~eexpressions it follows that:
Re Lum(n)~ = Re ~tm(n)] ~ hmp(n) - Im [tm(n)] ~ hmq(n 5 . (64) ::.
Im rUm(n)~ = Re [tm(n)] ~ hmq(N) + Im ~tm(n)] ~ h (n The complete implementation of the signal :~
channel 4(m) is shown in Figure 28.
In chapter E(2~4) it is already described that the transformation-device 3 produces the real :~;
part Re [sm(n)] and the imaginary part Im L sm(n)~ ~l`
: of sm(n) on separate outputs. For processing this complex signal {Sm(n)} in the signal channel 4(m) : .
this signal channel comprises two auxiliary channels 4(m,1) and 4(m,2) to which the signals Re [~m(n)]
and Im tsm(n)] are applied respectively (see Fi-.gure 28). Each of these auxiliary channels comprises ~ :*
an SRI-element 5(m,1) and 5(m,2) respectively at the -:
outputs of which the real part Re ~t~n(Il)~ and the imaginary part Im [tm(n)] of the complex signal ~ -[tm(n)~ are produced respectivaly, Said last outputs are furthermore connected to inputs of the digital filter 6(m). In general this filter com-prises four reali~able digital filters 6(m,1), .
6(m,2), 6(m,3) and 6(m,4) having the transfer ~ 59 --~ '' ' 10.6.77 S Hmp(~ )~ Hm~( W), _ Hmq(~) and H (~) respectively As shown in the Figure the inputs of ~
these filters are connected to the outputs of the -SRI-elements 5(m,1) and 5(m,Z). The outputs of the filters 6(m,.) are connected to inputs of two adders 6(m,5) and 6(m,6) whose outputs 6(m,7) and 6(m,8) -~
constitute the outputs of the digital filter 6(m).
At the output 6(m,7) the real signal Re [um(n) 3 is produced and at the output 6(m,8) the real sig-nal Im [um(n)] , which signals represent the real ~' and the imaginary part respectively of the signal ~ ~ (n)~ .
- E(2.5.2) The transfer matrices Hp(~J ) and Hq(~J ) To determine the transfer functions ~ Hm ( ~) and Hmq(~J) we wlll start, for simpIicity, from the expressions (48) and (49) wherein the am-plitude and phase distortion factor ~ J ) will be assumed to be equal to unity. At the same time, -it will be assumed that the FDM-signal is formed ~
. ~ . .
by the signal Re [v(n) ] . For the cases ~ot con-sidered here, which are indicated in chapter F(2.3), what follows here below proceeds in a similar manner.
In the first place the transfer matrices H (~ ) and H (~J ? wil 1 be determined for the ;
TDM-FDM arrangements with 4 1 ~ - for which the FDM-condition (49) applies. It now follows from -(61) and (49) that:
- 60 - ~;~
10.6.77 ',,:
_ (~J ) = J Hp(~U ) so that: ~-H (~J ) = ~(A(N)T) 1 Hp(2MT - ~Jo) = Hp(~ 0)~= ~(A(N)T ) (65) ;
~q(~Jo) = 2j (A
H (2M~ -~J ) = 1 (A~N)T~)~1 - From (65) it follows that the transfer function Hmq(~J) with 0 ~ ~J < 2 ~ represents the Hilbert transform of H p(W ). The resultlng implementation ~ of the signal channel is shown in Figu~e 28a. It should be noted that in (65) the transformation matrix A is assumed, in general, to be complex. ;
If, this matrix A ~s real then the implementation r of the signal chalmel remains, however, the same ;~ 15 as that shown in Figure 28a. It should also be not-ed that the matrix A(N)T is not assumed to be -~
singular, ~
In the second place the transfer matrixes ~--H (~ 0) and H (~J ) will be determined f~ the FDM~
arrangement with ~J1 = , for which the FDM-con-dition (48) applies. This FDM-condition (48) re-presents an equation which for a given transform ,.
~- ~
... . ' ' ~ . . .. . . ~ . ! . . ' ' , PHN 87~1 ~ 3,~ 10.6.77 '~
~, matrix A( ) comprises two unknown transfer matrixes.
To be able to dete.rmine these transfer matrices un- ~-ambiguously suitably chosen additional conditions can be imposed on either the transfer matrices or the transformer matrix. In what follows hereinafter three possible additional conditions will be mentioned by -way of example.
1. A first additional condition is, for example, .
H(~Jo) = H (2MT _u~O) (66j From (60) it then follows that:
H ( ~0) = 0 ~ .
with (66) the FDM-condition (48) changes into:
(N)T ~ A(N)~T).H (~J ) 5 2IN
1~ If the transformer matrix is further given by:
A(N) = Re [A(N)~ + j Im [A(N)]
then it holds that: .~:
p(~J0) = ~Re [A( ) ~J ;~
(67) and that: H (2~1T ~ (Re ~A(N)T]) Herewith it is assumed that the matrix Re [A(N)T~ ;~
_ 62 -. .
` `\ :
PHN 8731 ;
1~3~ 10.6.77 ; :.
is not ~ingular. The implementation of the signal channel resulting from this additional condition , ;~ follows from Figure 28 and is shown in Figure 28b. `~
2. As additional condition it might also be assumed that the transformer matrix comprises real elements only. This mean~ that:
Im LA ( N ) ] = O ( 68) A(N) A(N) ..
so that:
:: .
Im [sm(n)] = *:
mk =
- . .
:~; ` ~r The FDM-condition (48) now changes into: `~
(N)T , [H( W ) + H~(2~1T~ -~J0)~ = 2IN
With (60) it follows thab: ;
_p(~lO) = ~A(N)T]
Hp(2MT7r _ W ) = { ~A( ) ~ ~ (69) ~, H (~ ) = H (2MT ~ ~Jo) = arbitrary.
If this second additional condition is combined with the condition sub 1., that is to say if H (h~ ) = 0, then the signal channel shown in Figure 28 changes -`. ", ' .
,: .
~ 45~4 10.6.77 ~
into the signal channel shown in Figure 28c. ~~
3. A further additional condition is, for ;
example, H (2MT ~ ~Jo) = (70) ~ ~ -Herewith and with (65) the FDM-condition for ~J1 =
changes into:
A H(~Jo) = 2IN (71) The set of equations (70) and (71) is closely related to the set of equations defined in ~49) so that for this third additional condition :':, , a set of expressions related to (69) is applicable.
The signal channel is again implemented in the ', manner shown in Figure 28a.
. .
; From the above it follows that the FDM-~15 condition (49) for ~J1 ~ may be considered as a special case of the FDM-condition (48) for ~1 = ' This can also be interpreted as follows: if the transfer function Hm( ~) of the digital filter 6(m) with m = 1, 2, 3, ....N is chosen such that the modulus of Hm(hJ) is equal to zero, that is to sa~ -¦ Hm(~ = O in the range M T ~ ~)~ 2M T a fre~
quency shift ~) 1 is still pernlissible for the FDM- -signal within its fundamental period provided M is -taken greater than N(cf. the Figures 20b and 20a). ;~
It should be noted that with the help ~ - 64 . ~
10.6.77 3'~ ;-of the general theory which is available therefore (see for example reference 14), a digital filter having a given transfer function car. always be im-plemented. Consequently, in what fol~ows hereinafter ~~
the specific implementation of a digital filter hav-ing a given transfer function will not be discussed. -~-E(?.6) Simpl fication of the TDM-FDM arran~ement.
In the transformation device 3 of the TDM-FDM arrangement shown in Figure 21,N multipli-cation factors amk are used. As has already been remarked the complexity of the TDM-FDM arrangement ` ~ -is determined inter alia by the value of N. This -complexity is also determined by the value of the - -sampling rate T with which digital signal compo-nents tm(n) are applied to the digital filter 6(m).
This sampling rate determines the complexity of this digital filter. For this reason, if UV1 = O
the inorease factor M is taken equal to N.
If N is even a considerable simplification ;~ -of the TDM-FDM arrangement can yet be ~btained in the manner shown symbolically in Figure 29, This TDM-FDM arrangement comprises three TDM-FDM sub-arrangements 17, 18 and 19. Each of these TDM-FDM ~-sub-arrangements is implemented in the manner which is identical to the TDM-FDM arrangement shown in Figure 21. However, the sub-arrangements 17 and 18 are N-point TDM-FDM arrangements and the sub-- 65 - ~ -. . .
:::
::
P~IN 8731 :
10.6.77 .",, ~
arrangement 19 is a two-point TDM-FDM arrangement. As shown in Figure 29 the components of the input signals fXk(n)~ with k = 1, 2, 3- ...2 are supplied to the sub-arrangement 17 and those of the input signals : ;
~xk(n)~ with k = N ~ 1, 2 + 2, .... , N to the sub-arrangement 19. These sub-arrangements 17 and 18 : : produce the digital FDM-`signals f y l (n ~ and f Y2 ( n)}
respectively, these signals having associated there- I ~
with a sampling rate N2T if M=N. These digital slgnals i . ~-.
f Y1(n)} and {Y2(n)~ are applied to the sub-ar-rangement 19 which produces the desired digital FDM- :-` .
ignal of the N input signals {Xk(n)~ k = 1, 2, ... ~ ~-The total number of multiplications to be perfor~led .
15: in the three transformation devices in one period . ;
T now only amounts to 2. (N2)2 + 2 . 22 = N2 + 2N. ~
In addition, the sampling rate of the digital sig- ;;~ .
; nals which are applied to the digital filters in the sub-arrangements 8 and 9 is now equal to 2T~ so that ;~ 20 considerably less calculations per unit of time need :;~
be performed in these digital filters then in the di- I ~ -gital filters which must be used in the TDM-FDM ar- I :~
rangement of Figure 18 in the case M = N and ~ ~1 = O. ' ' I :"'~
If N = 2~ , wherein ~ represents an in- 1 ~:
teger, eac~l of the TDM-FDM arrangements 17 and 18 .
can in itself be constructed in a manner sho~l in .
_ 66 - . ~ ;
.
10.~-77 i~4~ 4 ~:
Figure 29.
It should be noted that the implementation ~of the TDM-FDM arrangement shown in Figure 29 can also be utilized for converting complex signals in-~ to an FDM format. As already indicated in chapter E(1.4) a real slgnal f xk(n)~ can be converted by means of the complex modulator shown in Figure 17 r into a complex signal which i9 composed of two -real signals Re ~ Pk(n)~ and Im [Pk(n)3 with, for example, the frequency spectrum shown in Fi-gure 18b and 18c respectively. To be able to con- ~vert the complex signals into, for example, the ;~ , SSB-FDM signal shown in Figure 20a the signals Re [ Pk(n)] are applied to the sub-arrangement ~15 17 shown in Figure 29 and the signals Im rPk(n)]
are applied to *he sub-arrangement 18. The FDM-signals fy1(n)~ and ~Y2(n)~ thus obtained are now not applied to the two-point-TDM-FDM arrange-ment 19 but are added together.
~(2.7)The TDM-FDM arranFement usin~ a fast trans-formation-device.
As indicated in chapter E(2.5.2) the FDM-condition for UVl ~ O must be considered to be a special case of the FDM-condition for W 1 = - As in case W 1~ the increase factor M of the SRI-elements must be greater than N it will be assum-ed hereinafter that ~ 1 = and M = N.
. .
PHN ~731 10.G.77 $~534 -~ ~
, If in the TDM-FDM arrangement of Figure 21 ~ the number of input channels N is equal to 2 - wherein ~ represents an integer and if the matrix A is chosen such that it results in a fast imple-mentation then not only the number of multiplications which must be performed in the transformatiGn-device is drastically reduced in this TDM-FDM arrangement but the digital filter 6(m), m = 1, 2,.~.. N can be ~
considerably simplified. ~ -Figure 30 shows a TDM-FDM arrangement for N = 8 wherein the matrix A(8) on which the trans- ; , formation device 3 is based satisfies the above-mentioned property and thus enables, for example, the fast implementation shown in Figure 27 with ; two-point transformers. In this Figure 30 elements which correspond with Figure 21 and Figure 27 have been given the same reference numerals. This TDM~
FDM arrangement is provided with eight input chan-nels 1(k) with k - 1, 2, 3, ..... N(=8). The input signal9 {xq(n)3 with q = 1, 2, ....... . . . N(=8) are applied to these input channels in an appa- ~-;
rently arbitrary sequence. The sequence of these input signals is chosen such that an FDM-signal is obtained having exactly the same shape as the FDM-signal which is produced by the arrangement of Fi-gure 21 (c~. Figure 2~b). As in Figure 21, the in-put channels to which the input signal.s with even ;' ~
\
3 4 10 6-~7 , ~ .
number (q = even) are .applied are each provided with a sideband interchanging.modulator 2(p) with p =
1,2~3~4. These input channels are connected to the inputs of a fast transformation device 3 which is -eonstructed in the manner shown:in Figure 27 and :~
which produces the complex output signals ~sm(n)~
with m = 1, 2, .... N(=8). These output signals are .
applied to N signal channels. Each signal channel eomprising means for increasing the sampling rate . ~.
: 10 by a desired factor. Contrary to the signal chan- -: :
nels of Figure 21 the signal channeIs of Figure 30 ~ -. .
: are partly used in common, so that each signal eharmel may be eonsidered as having been built up -~
.:
from a plurality of sub-charmels. In the Figure the ~fir~st channels are indicated by 21(.), the seeond sub-eharmels by 22(.) and the third sub-channels by 23(.). The signal eharmel w~eh in Figure 21 is in- ;:~
dieated by 4(1) is now formed in the arrangement of Figure 30 by the series-connected sub-ehannels ;.~.
21(1), 22(1) and 23(1). Likewise, for example the ;.
signal ehannel which in ~igure 21 is indicated by 4(2) is now constituted by the series-eonneeted sub~ehannels 21(2), 22(1) and 23(1).
. ~ : : .
The sub-ehannels 21(.) of Figure 30 each eomprise a series arrangement of an SRI-element 24(.) and a digital filter 25~.). The sub-channels 22(.) eaeh comprise a series arrangement of an adder - 69 - :
~..
10.6.77 :~
1~14~3~ :
~,.
26(.), an SRI-element 27(.) and a digital filter 28(.).
The sub-channels 23(.) each comprise a series arrange-ment of an adder 29(.~, an SRI-element 30(.) and a digital filter 31(.). In addition the sub-channels ;
23(,) are connected to inputs of an adder 32 which produces the desired digital FDM-signal ~y(n)~
Because the transformation device consists of two-point transformers the increase factor of all SRI- :
elements is equal to 2. ~.
In Figure 30 the transfer functions of the various digital filters are indicated by H(1)(~
N(12) (~J), H21)(~), H212)( W) and so on. In general ~`
such a transfer function will be indicated by Hiz)(~
-: -~ hereinafter wherein i = 1,2,3,..7; j = 1,2; and ~ :
- .
wherein z represents the number of the sub-chan-nel z = 1,2,3.
~ecause the digltal signals which at least are applied to the digital filters 25(.) and 28(.) occur with a considerably lower sampling rate then :
the digital signals which are applied to the digi.-.-tal filters 6(.) of the arrangement of Figure 21 ;
the transfer functions of the first-mentioned fil- ; .~
ters can be realized in a considerably simpler man- . .-ner.
In order to obtain the desired ~DM-signal at the output of the adder 32 the transfer functions of the signal channels and the matrix A( ), whose 10,6.77 $$~i34 ~ :
..;~
fast implementation is sho~n in Figure 30 must satis- ~
fy the FDM-condition (41). : -The transfer function of a signal cha.nnel ::~
is now given by the product of the transfer function ~ ~
of the various digital filters which are found in ~:
the successive sub-channels which together constitute the relevant signal channel- So, the transfer func-tion of the first signal channel is, for example, equal to:
Hl(W ) = H1~ J) H(1) (~).H(3)( ~) that of the seoond signal channel:
2(~J) = H(1)(~J ) 'H(1)(~J) (1) that of the third signal channel: .
H (~) = H(1)(~J) . H(2)(~) .H73 ( W).
.15 The fundamental interval of these transfer fu.netions H1(~V), H2(~J), H3(W )... H8(W ) is equal to 8. 2T . If for these transfer funetions, in ae- -eordance with expression (46), the matrix H(~o) is defined again then the FDM-eondition (48) must again be satisfied (M=8).
In the embodiment shown in Figure 30 this FDM-eondition is satisfied by having the transfer-funetions Hiz) (~ ) and Hiz)(~) as well as the ~14534 PHN 8731 '` ,' matrix Aiz)satisfy the FDM-condition. To be able to indi- -cate this more concretely we define the filter sub-matrices~
~ Hiz)(~z) Hil)(~ +2Z ~
Hiz(~z) ( 2 ~ (72) ~.
1z ( z) H(2)(~ +2Z 12~
H ( l ) ( 2 Z+l ~ _ ~ ) H~ ( 2 Z+1- 2z )~ -~Jz ] "~'' : ''''` ~;
Hiz(2Z+l ~ -~z)= ( ) \H(2)(2Z+1 -~ ) Hi2)[(2Z+1-2Z l)~r-~z]/
- .... . .
wherein i = 1, 2, 3, ... , 7; z = 1, 2, 3. .. ~
10~ 2Z-l ~ ,~ ",~,, z T `` ~
` ~ . :
T fsz Herein fsz represents the sampling rate associated with the input signal of the digital filter in the zth sub-channel and :~ .
which has the transfer function Hiz)(~J). Again T represents ~.
15 the sampling rate associated with the signals {sm(n)}. So it .
holds for the filters 25(.) that z = 1 and, consequently, :`-fsz = fSl = T. It likewise holds for the filters 28(.) that z = 2 and for the filters 31 (.) that z=3. ~;
By means of the filter sub-matrices (72) :
~ ! i - 72 - ~
' ~ , ~4~ 10.6.77 ~ ' ; and (73) the FDM condition (48) now changes into:
~iz Hiz(~Jz) + (A(z~ iz(2 ~T ~ ~Jz) =~2I2 (74) The sub-channels 21(.), 22(.) and 23(.) are now all implemented in the manner as indicated in Figure 28. The index m which is used in Figure 28 for the denomination of the transfer functions must now be replaced by a combination of indices i, z and ~. This means that the digital filter HiJ)(~) is now built-up by means of the digital filters HiJ)( W) and Hi3)(~) which are given by (603. For the additional conditions specified in chapter E(2.5.2) the general implementation indi-cated in Fi~ure 28 of the sub-channel changes into ~ that of Figure 28a or 28b or 28c.
1~ It should be noted that in the embodiment ;
shown in Figure 30 the fast implementation of the transformation device was based on two-point trans- -formers each having associated therewith a 2x2-matrix Ai2). If~ however, this fast implementation was based on four-point transformers each having associated therewith a 4x4-matrix Ai4) then the above holds similarly.
E(? 8~ Transformer matrices and transfer functions.
From the above it will be clear that a great number of transformer matrices is suitable in ~ :
principle to sérve as the base for the transformation device. However, not all transformer matrices which . . .
10.6.77 ~L4~3~ :
:,~
,.
~,.
are suitable in principle will result in a realizable TDM-FDM arrangement because they will lead to an ex-cessive number of multiplioations in the transforma-tion device or ts very complicated digital filters.
In this chapter a number of matrices will :~
be described by way of example, which may result in simple digital filters as well as in a simple trans-formation device and which will allow a fast imple-mentation of the transformation device.
A class of matrices which satisfies the `~
above requirements is formed by the Hadamard matrices `
; ~ where ~ = 1,2,3,.... which are defined as fol- -lows:
.: .
:: ~ .:.
, ~ . .... .. . . .. _ ... _.; .. _ __ . .. ..... ........ .... . .. . ..
J~ -1 ~V ~ . ~ V =1' . .~ _ _ :: .
:, ~:
wherein ~ = 2~ 3~ ... and where ', ,...: :-~1 1 , ,.
01 = ~ (76) -:' A Hadamard matrix is a real matrix which satis-fies (68).
Because a Hadamard matrix allows a fast implementation only the simplificat~.ons will be indicated here which are possible in the TDM-FDM
, P}IN 873i 10,6.77 1~4534 ; - ~
', . :~` ' arrangements shown in Figure 30. : -In the arrangement of Figure 30 the number :~ -~: , - . .
:: of input signals N is equal to 8 = 23. This méans . . ; ~
that the transformation device 3 must be based on - :
the 8x8 Hadamard matrix 03. This results in that:~
Aiz) = 01 for all i and all z (77) :~
. Now as an additional condi.tionit can be chosen: ;
; . Hi ( W ) = ~iz(2 T ~ W
' 80 that Hizq( z) ~ :.
;~ : . . , ~ ~ . From (67) it now follows that: :
~ ~ HiZP(~JZ) = L01T] = 2 (1 1) for all i and all z ~;~ N (zzt1 ~r ~ ) _ ~ ~ T~ -1} (78) 2 (1 -1) for all i and all z : ':
' From (76) and from Figure 23 it follows that the two-point transformers which together con- ~:
, stitute the transformation device of Figure 30 are I ;-I ,.
very simple. Because the multiplication factors ~ :
indicated in Figure 23 are now given by: ¦ ;
~,, .
: ~ 75 10.6.77 534 .
,..~
: ~ 11 = ~12 = ~21 =
22 = -1 12 = ~21 = ~ 22 =
the multipliers 9(1), 10(1) and 9(3) o~ Figure 23 -~
oan be replaced by through-colLnections~
~: : Before the influence of the Hadamard matrix ..
:
on the transfer functions of the digital ~ilte~ is --`, ~: described it is be noted that the TDM-FDM arrange-; ment shown in Figure 30~comprises seven circuit 10~ configurations of the type depicted in Figure 31. .
This circuit configuration comprises two channels ~ .
I and II. To each of these channels a digital sig- ~
nal having associated therewith a sampling period T is applied and this configuration produces a : .-: : 1S digital output signal having associated there- : :
!- - :
with a sampling period Tz, For the sub-channels ~ .' . ~ 21(.) in Figure 30 z = 1, for the sub-channels . 22~.) z = 2 and for the sub-channels 23(.) z = 3. ~--By choosing .the Hadamard matrix as the ~.
transformer matrix it is now obtained that: :
~: -:
~ Hiz(~ ) = H (~V
.
so tha.t the circuit configuration shown in Figure ~ -;
31 changes into the circuit configuration depicted in Figure 32 (see also Flgure 28c). .
"
. - 76 .~ . .
:~:
\
5 ~ ~ 10-6-77 1. ' .
, From (78) it fo~ws that:
~( )( W ) = H(lP ~JZ~Z T ) = 1 H(l) (2Z~ W ) = H(1) [(ZZ~l ZZ-~
1.
so that the digital filters having the transfer func-tions Hiz)(w ) are each equivalent to a through- 1 connection. For completeness this situation is de-picted in Fig. 33. Consequently the TDM-FDM arrange-ment comprises only N-1=7 digital filters.
;10 From (78) it furthermore follows that ~or ! ` -~
the digital filters having the transfer functions ~ ~
Hizp(~V) it holds that: ;
Hizp ( W z ) = H ( 2 ) ( 2 ~+1 '17' izp Z T ) izp ~ ) T -~ ~z]
i =-1.
This means that these digital filters are formed by all-pass filter which introduce a phase shift of 0, - ~ or ~ ~ . Figure 34 shows, for completeness, the argument of H(2) (~J) as a function of ~
A further simplification of the circuit configuration of Figure 33 is depicted in Figure 35. This configuration differs from tllat of Figure ,. ~ 10.6.77 $;~ `
..
33 in that the channel II is now constituted by a se-ries arrangement of a digital filter having a trans-fer function Gz,- the SRI-element having an increase factor 2 and a delay device having a delay time zZ .
The amplitude-frequency characteristic of both the ~ r ; digital f'ilter and the delay device is equal to -unity. Figure 36 shows two fundamental intervals ;
of the phase-frequency characteristic of the digi~
tal filter. For completeness Figure 37 shows the phase-frequency characteristic of the delay device s~
over one fundamental interval. The phase-frequency chzracteristic shown in Figure 34 is now obtained by adding the characteristic shown in the Figures ~ ;
36 and 37 together. ~ -In a practical embodiment the circuit con-figuration Or Figure 35 can be reduced to the confi-guration depicted in Figure 38 and which is equi-valent to the configuration depicted in Figure 35 and which is also provided with two channels I and - ~
. ~ ,. .
II. The functions of the SRI-elementsJ the delay device and the adder shown in Figure 35, however~
are now concentrated in a switching device 33. This switching device, which is only symbolically shown, ;
is controlled by switching pulses which are produc- ;
ed with a period z. At the occurrence of the first of two successive switching pulses the channel I ;`
is connected to the output 34 of this switching : .
PlIN 8 7 31 10.6.77 ~$4LS34 - ,,, device and when the second switching pulse occurs the ~.
channel II is connected to this output 34. At the output 34 of the circuit configuration a digital signal is produced having associated therewith a .' sampling period z and;which is'formed by a suc- ' ' - cession of signal components which alternatingly ~' originate from the channel T and the channel II :' (interleaving). ' ;-For completeness Figure 39 shows the-~com- ~ ' plete implementation of the TDM-FDM arrangement ~.
wherein the Hadamard matrix Ai )= 01~ defined in'' ~: (76~, is the basis of each of the transformers.
-This'TDM-FDM arrangement follows., together with ' the preceding description, from the Figures 30, 23, .' -~
15 ' ' 27'and 38. The'digital filters, having the trans-fer functions G1, G2, G3 are all all-pass fiiters having the phase-frequency characteristicsshown in' ~igure 36. ' :~
A further real matrix which results in '~ ' 20 both simple digital filters and a simple transforma- ` .~ :
tion device and which allows a fast implementation of this transformation device is obtained if ~.
. ~ 1 0\
Aiz2) = ~ ¦ for all i and all z (79) .
~_1 1/ :
If it is again assumed that: Hi ~ ) = O it then follows together with (74) that:
~ 79 -,~.
! PHN 8731 10.6.77 '-~
I
- /~ 1 : ~
-iz(~Jz) = Hizp( Z) 1 for all i and all z :~
0 1 ' (80) H (2Z+l T~ ~ ~) ) = HiZp(2æ+l~T ~~Z) = ( ~ ) '~
; ~ 1 ~
Also with this choioe of the transformer matrlx ~,' '-;
Aiz the circuit configuration of Figure 33 applies, so that the TDM-FDM arrangement of Figure 30 can !
again,be realized with only N-1 digital filters.
. ,~
. The digital filter shown in Figure 33 '.- :
, with transfer function Hi2z)(~J) can now again be ,~:, realized by a cascade arrangement of a digital fil- ~
ter with transfer function Dz(h~) and a delay de~ice ".,'.~.
having a delay time Tz (see Figure 40). The amplitude- ,':'.
frequency characteristic of Dz(W ) is shown in Figure ~
41 and its phase-frequency characteristic is shown `~ ,., in ~igure 42.
' ~ When the transformer matrix is chosen ac-cording to (79) and when a phase distortion may be allowed, the circuit configuration of Figure 33 can also be implemented in a manner different from the implementation which is depicted in Figure 40. Tak- ~ -ing into account the remarks indicated in chapter , E(2.3), the FDM-conditlon (74) CQn now be written in the form: .
.
, - 80 - ~ `
1~45~ lo. 6.77 ~.
", ) Hiz (W ) = 2I2 diag ~ ~iz(~ z)] ~ ;
~' so that:
Hizp(h/ z) = ( )(,j~iz; 1 ( ~z) 0 O 1 0 yiz;2(~z) 1 .
iz;1(~z) ~iz;2( ~ z)~
. :, . ~iz;2(~ z) ::
Because Hizp(~J ) = O a phase-frequency function ~ ;
may be allotted to Hi2)(~Jz) which is equal to yi 1(~Z) If it is now assumed:
; izp ¦ z(~ )¦ ~iz,2( W ) whereby the f~mction¦D~ is depicted in Figure . I
41~ then we are stlll free to choose ~ iz;1(~z) equal to ~ iz;2(~Jz)' This means that in ~igure 33 a digital ~ ter having the amplitude-frequency :.
function shown in Figure 4t and a phase-frequency function which is given by ~i ~2( W) must be in-cluded in channel II and that a digital all-pass filter having the saMe phase-frequency function ~iz;2(W ) must be included in the ch~nel I.
In chapter E(2.4) it was noted that I :
a fast transformation device can also be implemented i -~
' 81 - - ! :
, . . I
.~ . . .. . . ...
1$~3g 10.6.77 ~ -:~ -: - -, by means of, for example, four-poin.t- transformers. j : :
Such an implementation can,. for example, be advan- ~ ::
tageous if N=2 wherein ~ represents a positive integer, With such an implementation the increase ~ -.
factor of SRI-elements in the various sub-channels . --of the device of Figure 30 is equal to four and ~ ~
the transfer matrix i9 defined for four sub-channels . ;:
of the same number. For completeness a real trans- i~
former matrix will now be given which might be the -::
~ ::
; lO basis for the four-point transformers. Such a matrix, ~:; for example, is the real matrix .~ :;
/ 1 1 o : ~ A(4)= 0 1 -1 0 for all i and all iz O 0 1 -1 z : `.
150 0 0 1 .
If now a8ain the additional condition is given by:
then it follows from (69) that:
Hi ( W ) = (Ai4)T) ? O so that:
11 0 , O - O \
--izp(~'~
~ - 82 t `': " ' ;~
PHN (~731 ~ 4 1o. 6.77 In chapter E(2.7) it is indicated that, the TDM-FDM-arrangement depicted in Figure 21, can be , replaced by the TDM-FDM -arrangement depicted in Figure 30, when the transformer matrix allows a fast implementation. In case W 1= and the transformation - device is based upon a real Hadamard matrix the ar-rangement of Figure 30 can be still further modified to the arrangement shown in Figure 39.
If ~J1 ~ the signals {rk(n)} applied to the transf~mation device in the arrangement of Figure 30 are complex. This means that the increase factor of the SRI-elements in the various sub-chan-nels must at least be equal to three. From Figure ; 30 it will be clear that this means that if with bhe arrangenentof Figure 30 eight signals {xk(n)}
which are all unequal to zero must be converted in-to a FDM-signal a sampling rate must be associated with the output signal {y(n)} which is at least equal to 27/T. If this is compared with the sam- -plin~ rate whic~ is associated with thesignal f y(n)} which is produced by the arrangement of Figure 21 and which is at least equal to 9/T
then it appears that the implementation shown in Figure 30 is not attractive if hJ1 ~ -If for special circumstances, for example, if t~le signals` frk(n)3 represent complex signals, in the arrangement shown in Figure 30, a complex -;
:
- ~ r PH.N 8731 10.6.77 ~-~$~
transformer-matrix is preferred over a real transfor-mer-matrix then it would, for example, be possible . to take therefore the ma~rix~
/l~j l-j\ .
Al2) = 1 ~ ) for all i and all z ;~
l-j -l~J , To indicate the influence of this choice of the trans- ~`
former-matrix on the digital filters we shall again restrict ourselves to the case ~J1 = 0. I~ the ad-ditional condltion:
Hiz(2 T ~ ~Jz) =
is now chosen then the sub-channels must be construct-ed in the manner shown in Figure 28a. From (65) it now follow~ that: ~-/1 l\
HizP(4J z) = 2 ( ) for all i and all z _,~ , . , --i Zp ( 2 1 ~ .
As regards the transformation device the above means that~
_ 84 10.6.77 ;
3~ ;
. ~ .
~11 = 1- ~ 12 = 1 12 = -1 21 = 1 CX 22 = -1 `
~ 2~ 22 = 1 5 . So it applies for the digital filters HiJp(~ ) that the filters having the transfer function Hizp(w ) ! ;~
just form through-connections and that the digital filter having the transfer function Hi2)(~J) forms an all-pass filter which introduces a ~ 2~ phase .
10 ~ shift in the interval 2 l~r ~ ~ C 2Z 1r and which introduces a + 2 phase shift in the interval T ~ ~2 i ~1T . For completeness ~igure 43 .
shows the argument of Hi,2,)(~) as a functioll of~J .
: .As has already been remarked, if N = 4 , j:~
15 . wherein ~ represents a positive integer, the trans- ;~
formation device can also be based upon four~point transformers .
A possible 4x4-matrix is the complex matrix !
1 1 1 1\
Aiz ~ ~ with J ~ (81) - j - 1 j `''' for all i and all z.
With the additional condition: ./
Hiz(2 ~ _kJ ) = o ~:
' - 85 - .
~ 4 10-6-77 ~, ;
, ~ .
: , .
~ it follows from (65) that:
/1 1 1 1 . ~ . , (~U ) ~ with j = ~ (82) for all i and all z.
In full agreement with the above the transfer functions i1 ~ ;, in the various intervals follow from these transfer- j matrices.
For completeness Figure 44 shows an elaborat-~~ ed embodiment of the arrangement of Figure 21 wherein the real signals ~xk(n)} are converted into complex signals by means of the complex modulators. In the arrangement: of Figure 44 N = 4, M = 5 and the transformation device is based upon the matrix Aiz) given in (81). For the transfer functions of the digital filtqrs it is again assumed that Hm (5~ J0) = 0, so that the signal channels must be implemented in the manner shown in Figure ~ I
28a. The transfer functions Hmp(~) of the digital , ~ -filters incorporated in these signal channels are ~ -given by (82). The amplitude-frequency flmction is the same for all digital filters Hmp( ~ ) with m = 1,2, 3,4 and is shown in Figure 45.,'rhe phase-frequency functions of the various digital filters H p are depicted in Figure 46, '~, .
_ ~6 , 1~3L4S34 1 o . 6 . 77 .'; :, ` As can be easily checked, taking into ac-count the expressions (16), the TDM-FDM arrangement shown in Figure 44 produces the FDM-signal shown in ;
Figure ZOa.
It should be noted that in the arrangement of Figure 44 only the real part of the complex sig- -nals ~um(n)} is determined. For, the imaginary - ~
part of this ~ignal has no oontribution to the de- ~-~ "
sired FDM-signal. 7 E(2.9) General remark~about the TDM-FDM arran~ement Each of the Figures 21, Z8, 28a, 28b, 28c, ~ ;
.
30,31,32,33,40 and 44 show series arrangements of an SR} element and a digital filter. In a practical em-bodiment of such a series arrangement the function ~ ~ Or the SRI-eIement and that of the digital filter are interwoven so that a practical embodiment o~
uoh a series arrangement is constituted by an in-terpolating digital filter which is also called a :, ~ ~ sampling rate increasing digital filter. For the ~ -; 20 implementation of such a digital filter we refer, rOr example, to references 15, 16 and 17. ~
2, The elements 8 shown in the Figures 21 ``
and 30 are used for mathematical purposes only. As ~
. .
appears from the Figures 39 and 44 such an element -~
is not used in a practical embodiment of the TDM-FDM convertor because the real and the imaginary ~
part of a complex signal are available separately. ~ -~ ~' PH~ 8731 10,6.77 , To determine the signal Re L v(n)~ it is sufficient to apply the signals Re ~ um(n)] to the adder 7 and 32 respectively.
3. In the preceding it was assumed that, if W 1 = the channel signals in the frequency spec~
trum of the FDM-signal are located as shown in Fi-gure 20b. This means that for N = 4 the FDM-signal of the four baseband signals ~ x1(n)~ ' {X2(n)}
~x3(n)} and ~x4(n)~ is located in the frequency band of 0< ~ 8T/ . However, if it is desired to have the FDM-signal situated, for example, the fre-quency band ~C 9 ~ then, of course, ~J1 may be ~
taken equal to T . It is, however, simpler to choose ~ `
N equal to 5 and to compose a FDM-signal starting ~ -from the five baseband signals ~ xO(n)} ~ {x1(n)~
2(n)3 ~ {X3(n)~ and fx4(n)3 ~ whereby fxO(n)~ is equal to zero for all n.
4. If ~V 1 ~ ~ as in the TDM-FDM arrangement of Flgure 44 the transfer functions Hm(~J) must be equal to zero in those frequency intervals which are not covered by the FDM-¢ondition. --F(1~ The FDM-TDM arranFement.
Chapter E extensively describes arrange-ments for converting a plurality of discrete base-band signals into a discrete baseband single-side-band frequency-multiplex sign~l. This chapter briefly deals with arrangements formed by the trans-' . - 8~ -PH~ 8731 ~4~3~ 10.6.77 .
posed configurations of the arrangements described .
in chapter E and which thus convert a discrete base-division band single-sideband frequenc ~ multiplex signal into the original, spatially distributed, discrete base-band signals. The transposed configuration of a given arrangement is obtained by - reversing the direction of all signals;
- replacing the adders by branch nodes;
- replacing the branch nodes by adders;
- replacing the SRI-elements by SRR-elements;
- replacing the SRR-elements by SRI-elements. .
; The transposed configuration of the TDM-; ~: , , . ' .`' - FDM arrangement shown in ~igure Z1 is depicted in Figure 47. This FDM-TDM arrangement comprises a plurality of N signals ohannels 4'(m) with m = 1,2,3,N.
Applied to each of these signal channel~ is the FDM-signal fy~n)~ having the frequency spectrum y( W) which, for example, again has the form which for N- 4 is depicted in Figure 20a is applied to each signal ~:
channel. Each signal channel 4'(m) comprises a di- ~
gital filter 6~(m) having the transfer-function E (h~) ~ :
and an impulse response em(n), which, in general is complex em(n) - Re L em(n)~ + j Im [em(n)~
This di~ital filter 6'(m) produces a digital out-put signal t'~n(n) which is given by:
t'm(n) = Re ~ t m(n)] + J Im rt m( )~
PHN 8 7 31 ~:
34 1o. 6.77 If` ~y(n)~ represents a real signal it follows that (see 52)) ; [ m(n)~ = y(n) x Re ~e (n)~
: L [ ~ ::
The frequency spectrum T'~(J) of {t' (n)~ is given by:
Tm( W ) = y(~ Em~ ) . This signal ~t' (n)~ is applied to an SRR-element 5'(m). From (11) it follows that the components~
~10 5~m(n) of theoutput signal fs~ (n)3 of the SRR-element are equal to~
(n) = tm(Mn) From (12) it follow~ that the frequency spectrum S~ ) is given by~
Sm(~) c M ~ T'm ~+(q~1) wherein T represents the sampling period associated with the signal { s'm(n)~ :
The signals { s'm(n)} thus obtained are applied to the transformation device 3' which com- -prises N output channels 1(k) and whose operation can again be fully described by means of a matrix B = r bkm~ comprising the elements bkm. This trans-' _ 90 -PHN 8731 : -10.6.77 :
, formation device produces N digital signals -:
~k(n)~ , k = 1, 2, 3, ..... N, which are given l, .
by:
N . ~:
rk ( n ) = ~ bkm Sm ( ) ~ ~:
The associated frequency spectrum R'k(~) is given ~ :
by~
Rk(~J) = ~ bkm Sm( 4 ) , The complex signal {r'k(n)} is applied to a com- '- :
plex demodulator 1'(1,k) which produces the complex signal ~wk(n)} which is given by wk(n) = r'k(n)e j lnT.
Of this signal the arrangement 8(k) now produces the -real part of the signals fwk(n)3 . Th~ the out-put signal vk(n) of this arrangement 8(k) is given by v (n) = Re rwk(n)~ = ~ rwk( ) k Herein wk ~(n) again represents the complex conjugate of wk(n). For the frequency spectrum Vk(~J) of fvk(n)~ lt again holds (see (30)) ~ Vk(~V) = 2 fRk(~J + W 1) + ~ [ T ~~ +~J1]~
For the output channels l(k) with odd nwnber k it again holds that xk(n) = vk(n) and each the . .
output channels with even num'~er k again comprises ~ :
.
PHN 8731 - :~
1~4S3~ 1 o .~ . 77 '~
'`
a sideband interchanging modulator 2'(.) whose ,.
output signals constitute the desired signals k(n)} for k is even. ~ `
After some manipulation in the manner in-dicated in chapter E(2.3~ it is possible to derive the.TDM-conditions from the~above expressions. The .
. TDM-condition for ~J1 - O as follows: ~
. _ _ ,, .
-, . ..
m=I { km m ~Jo + (i-l) ~ ] 4 .'~
kmEm~ 0 ~ 1) T ¦} = S (83) :: ~ 10 The TDM condltion for ~Jl ~ is as follows: ;- "~:~
Em ~M T ~ WO + ~J1 + (i-1) T~ =
~ ~ ~84).
: ~ 1 N
~1 km m ~VO + ~)1 + (i-1) T-3 = ~ ki ::
In (83) and (84) ~ ki again represents the Kronecker symbol which is defined in (43). Furthermore it ap- -~15 plies that: :`
o~ w ,~
.-.
.i = 1, 2, 3, ... N.
Just as the FDM-conditions (41) and (42) the TDM con-' 10.6.77 ;
1~4534 :-ditions (83) and (84) can be written in the form of ¦
a matrix, the matrices defined in (46) and (47) be-ing applicable so that (83) changes into:
B(N) E(~J )+B(N) E (M.2 ~ _ ~U0) = 2NIN.diag ~ i(4Jo)~ (85) ~ '~
and (84) changes into:
E (M T ~ ~o) =
(86) ;
; B( ~ E(~lo) = 2NI~-diaF [ ~i`( ~ )~
Figure 47 shows a possible implementation ~ of a FDM-TDM arrangement wherein the relationship ;~ between the transfer functions Em( W ) of the digi-tal filters 6~(m), with m =1, 2, 3, ... N and the transformation matrix on whioh the transformati~n devioe 3' is based i9 given by the TDM-condition (84) or (86). As for the TDM-FDM arrangement a oom-plete freedom in the choice of the various matrices is possible so that what has been described for the ~-TDM-FDM arrangement fully applies for this -FDM-TDM ~ -arrangement.
Comparing the TDM-conditions with the FDM-oonditions shows that these conditions are not iden-tioal. In the oase the FDM-TDM arrangement is ob-tained by transposing the ~DM-FDM ~rrangement this .: ' , PI~N ~731 - ~
10.6.77 .-;: . .'"
is, however, only seemingly so. By transposing the ... .... ......... .... digital filters 6(m) Or Figure 21 these filters change into the digital filters 6'(m) of Figure 47.
It is true that this causes the implementation of . -the digital filter 6'(m) to be different from that of the filter ~6(m), but the transfer function of the filter 6(m) is not affected hereby (see reference 2) so that: . .
Em(~J) = Hm~J) . (87) It can be checked ln a simple manner that by trans~
: : posing the transformation de~ice 3 which i8 based on a matrix A( ) = ~ a k] the transformation de-. vice 3' is obtained which is based on a matrix A( ) = . .-[akm] so that 9(N) = A(N)T (88) ..
; Substitution of (87) and (88) in (86) again fur- .:
nishes the FDM-condition (49) and substitution of (87) and (88) in (85) furnishes the FDM-condition (48). From the preceding it consequently follows that a FDM-TDM arrangemant can be obtained by trans- `~
posing a given TDM-FDM arrangement or vice versa.
F(2~ General remarks about the FDM-TDM arrangement :: 1. Figure 47 and the arrangements which are obtained by transposing the TDM-FDM arrangements ::
shown in the Figures 30 and 39 each comprise series ~
_ 94 - , ,. .. l- .- - , ... .. . .
10.6.77 ' arrangements of a SRR-element and a digital filter.
In a practical implementation of such a series ar-rangement the function of the SRR-element a~d the function of the digital filter are interwoven. Such g a series arrangement is consequently implemented by means of an extrapolating digital filter, also called sampling rate reducing digital filter. For the implementation of such a digital filter we re-fer to the references 15 and 18.
2. The complex demodulator shown in Figure 47 can, as the further elements of this arrange-ment, be obtained by reversing, in Figure 17, the ~
- signal direction and by replacing the distribution ~ -point by a subtractor. With such ~ implementation of the complex demodulator the output signal here-of is given by Re [wk(n)3 so that the arrangement 8(k) can be dispensed with in a practical embodiment of the FDM-TDM arrangement.
3. The TDM-FDM arrangement and the FDM-TDM
arrangement need not form a system in which, for example, the TDM-FDM arrangement functions as a transmitter and the FDM-TDM arrangement as a re-ceiver or vice versa. Each of these arrangements can be used in existing PCM telecommunication sys-tems independent of the application of the other arrangement.
~ 95 , ~ .. ,.. . , .. . .. ,. . . .. ,,, !.'.. '.-. . .
these filters are connected to the outputs of the -SRI-elements 5(m,1) and 5(m,Z). The outputs of the filters 6(m,.) are connected to inputs of two adders 6(m,5) and 6(m,6) whose outputs 6(m,7) and 6(m,8) -~
constitute the outputs of the digital filter 6(m).
At the output 6(m,7) the real signal Re [um(n) 3 is produced and at the output 6(m,8) the real sig-nal Im [um(n)] , which signals represent the real ~' and the imaginary part respectively of the signal ~ ~ (n)~ .
- E(2.5.2) The transfer matrices Hp(~J ) and Hq(~J ) To determine the transfer functions ~ Hm ( ~) and Hmq(~J) we wlll start, for simpIicity, from the expressions (48) and (49) wherein the am-plitude and phase distortion factor ~ J ) will be assumed to be equal to unity. At the same time, -it will be assumed that the FDM-signal is formed ~
. ~ . .
by the signal Re [v(n) ] . For the cases ~ot con-sidered here, which are indicated in chapter F(2.3), what follows here below proceeds in a similar manner.
In the first place the transfer matrices H (~ ) and H (~J ? wil 1 be determined for the ;
TDM-FDM arrangements with 4 1 ~ - for which the FDM-condition (49) applies. It now follows from -(61) and (49) that:
- 60 - ~;~
10.6.77 ',,:
_ (~J ) = J Hp(~U ) so that: ~-H (~J ) = ~(A(N)T) 1 Hp(2MT - ~Jo) = Hp(~ 0)~= ~(A(N)T ) (65) ;
~q(~Jo) = 2j (A
H (2M~ -~J ) = 1 (A~N)T~)~1 - From (65) it follows that the transfer function Hmq(~J) with 0 ~ ~J < 2 ~ represents the Hilbert transform of H p(W ). The resultlng implementation ~ of the signal channel is shown in Figu~e 28a. It should be noted that in (65) the transformation matrix A is assumed, in general, to be complex. ;
If, this matrix A ~s real then the implementation r of the signal chalmel remains, however, the same ;~ 15 as that shown in Figure 28a. It should also be not-ed that the matrix A(N)T is not assumed to be -~
singular, ~
In the second place the transfer matrixes ~--H (~ 0) and H (~J ) will be determined f~ the FDM~
arrangement with ~J1 = , for which the FDM-con-dition (48) applies. This FDM-condition (48) re-presents an equation which for a given transform ,.
~- ~
... . ' ' ~ . . .. . . ~ . ! . . ' ' , PHN 87~1 ~ 3,~ 10.6.77 '~
~, matrix A( ) comprises two unknown transfer matrixes.
To be able to dete.rmine these transfer matrices un- ~-ambiguously suitably chosen additional conditions can be imposed on either the transfer matrices or the transformer matrix. In what follows hereinafter three possible additional conditions will be mentioned by -way of example.
1. A first additional condition is, for example, .
H(~Jo) = H (2MT _u~O) (66j From (60) it then follows that:
H ( ~0) = 0 ~ .
with (66) the FDM-condition (48) changes into:
(N)T ~ A(N)~T).H (~J ) 5 2IN
1~ If the transformer matrix is further given by:
A(N) = Re [A(N)~ + j Im [A(N)]
then it holds that: .~:
p(~J0) = ~Re [A( ) ~J ;~
(67) and that: H (2~1T ~ (Re ~A(N)T]) Herewith it is assumed that the matrix Re [A(N)T~ ;~
_ 62 -. .
` `\ :
PHN 8731 ;
1~3~ 10.6.77 ; :.
is not ~ingular. The implementation of the signal channel resulting from this additional condition , ;~ follows from Figure 28 and is shown in Figure 28b. `~
2. As additional condition it might also be assumed that the transformer matrix comprises real elements only. This mean~ that:
Im LA ( N ) ] = O ( 68) A(N) A(N) ..
so that:
:: .
Im [sm(n)] = *:
mk =
- . .
:~; ` ~r The FDM-condition (48) now changes into: `~
(N)T , [H( W ) + H~(2~1T~ -~J0)~ = 2IN
With (60) it follows thab: ;
_p(~lO) = ~A(N)T]
Hp(2MT7r _ W ) = { ~A( ) ~ ~ (69) ~, H (~ ) = H (2MT ~ ~Jo) = arbitrary.
If this second additional condition is combined with the condition sub 1., that is to say if H (h~ ) = 0, then the signal channel shown in Figure 28 changes -`. ", ' .
,: .
~ 45~4 10.6.77 ~
into the signal channel shown in Figure 28c. ~~
3. A further additional condition is, for ;
example, H (2MT ~ ~Jo) = (70) ~ ~ -Herewith and with (65) the FDM-condition for ~J1 =
changes into:
A H(~Jo) = 2IN (71) The set of equations (70) and (71) is closely related to the set of equations defined in ~49) so that for this third additional condition :':, , a set of expressions related to (69) is applicable.
The signal channel is again implemented in the ', manner shown in Figure 28a.
. .
; From the above it follows that the FDM-~15 condition (49) for ~J1 ~ may be considered as a special case of the FDM-condition (48) for ~1 = ' This can also be interpreted as follows: if the transfer function Hm( ~) of the digital filter 6(m) with m = 1, 2, 3, ....N is chosen such that the modulus of Hm(hJ) is equal to zero, that is to sa~ -¦ Hm(~ = O in the range M T ~ ~)~ 2M T a fre~
quency shift ~) 1 is still pernlissible for the FDM- -signal within its fundamental period provided M is -taken greater than N(cf. the Figures 20b and 20a). ;~
It should be noted that with the help ~ - 64 . ~
10.6.77 3'~ ;-of the general theory which is available therefore (see for example reference 14), a digital filter having a given transfer function car. always be im-plemented. Consequently, in what fol~ows hereinafter ~~
the specific implementation of a digital filter hav-ing a given transfer function will not be discussed. -~-E(?.6) Simpl fication of the TDM-FDM arran~ement.
In the transformation device 3 of the TDM-FDM arrangement shown in Figure 21,N multipli-cation factors amk are used. As has already been remarked the complexity of the TDM-FDM arrangement ` ~ -is determined inter alia by the value of N. This -complexity is also determined by the value of the - -sampling rate T with which digital signal compo-nents tm(n) are applied to the digital filter 6(m).
This sampling rate determines the complexity of this digital filter. For this reason, if UV1 = O
the inorease factor M is taken equal to N.
If N is even a considerable simplification ;~ -of the TDM-FDM arrangement can yet be ~btained in the manner shown symbolically in Figure 29, This TDM-FDM arrangement comprises three TDM-FDM sub-arrangements 17, 18 and 19. Each of these TDM-FDM ~-sub-arrangements is implemented in the manner which is identical to the TDM-FDM arrangement shown in Figure 21. However, the sub-arrangements 17 and 18 are N-point TDM-FDM arrangements and the sub-- 65 - ~ -. . .
:::
::
P~IN 8731 :
10.6.77 .",, ~
arrangement 19 is a two-point TDM-FDM arrangement. As shown in Figure 29 the components of the input signals fXk(n)~ with k = 1, 2, 3- ...2 are supplied to the sub-arrangement 17 and those of the input signals : ;
~xk(n)~ with k = N ~ 1, 2 + 2, .... , N to the sub-arrangement 19. These sub-arrangements 17 and 18 : : produce the digital FDM-`signals f y l (n ~ and f Y2 ( n)}
respectively, these signals having associated there- I ~
with a sampling rate N2T if M=N. These digital slgnals i . ~-.
f Y1(n)} and {Y2(n)~ are applied to the sub-ar-rangement 19 which produces the desired digital FDM- :-` .
ignal of the N input signals {Xk(n)~ k = 1, 2, ... ~ ~-The total number of multiplications to be perfor~led .
15: in the three transformation devices in one period . ;
T now only amounts to 2. (N2)2 + 2 . 22 = N2 + 2N. ~
In addition, the sampling rate of the digital sig- ;;~ .
; nals which are applied to the digital filters in the sub-arrangements 8 and 9 is now equal to 2T~ so that ;~ 20 considerably less calculations per unit of time need :;~
be performed in these digital filters then in the di- I ~ -gital filters which must be used in the TDM-FDM ar- I :~
rangement of Figure 18 in the case M = N and ~ ~1 = O. ' ' I :"'~
If N = 2~ , wherein ~ represents an in- 1 ~:
teger, eac~l of the TDM-FDM arrangements 17 and 18 .
can in itself be constructed in a manner sho~l in .
_ 66 - . ~ ;
.
10.~-77 i~4~ 4 ~:
Figure 29.
It should be noted that the implementation ~of the TDM-FDM arrangement shown in Figure 29 can also be utilized for converting complex signals in-~ to an FDM format. As already indicated in chapter E(1.4) a real slgnal f xk(n)~ can be converted by means of the complex modulator shown in Figure 17 r into a complex signal which i9 composed of two -real signals Re ~ Pk(n)~ and Im [Pk(n)3 with, for example, the frequency spectrum shown in Fi-gure 18b and 18c respectively. To be able to con- ~vert the complex signals into, for example, the ;~ , SSB-FDM signal shown in Figure 20a the signals Re [ Pk(n)] are applied to the sub-arrangement ~15 17 shown in Figure 29 and the signals Im rPk(n)]
are applied to *he sub-arrangement 18. The FDM-signals fy1(n)~ and ~Y2(n)~ thus obtained are now not applied to the two-point-TDM-FDM arrange-ment 19 but are added together.
~(2.7)The TDM-FDM arranFement usin~ a fast trans-formation-device.
As indicated in chapter E(2.5.2) the FDM-condition for UVl ~ O must be considered to be a special case of the FDM-condition for W 1 = - As in case W 1~ the increase factor M of the SRI-elements must be greater than N it will be assum-ed hereinafter that ~ 1 = and M = N.
. .
PHN ~731 10.G.77 $~534 -~ ~
, If in the TDM-FDM arrangement of Figure 21 ~ the number of input channels N is equal to 2 - wherein ~ represents an integer and if the matrix A is chosen such that it results in a fast imple-mentation then not only the number of multiplications which must be performed in the transformatiGn-device is drastically reduced in this TDM-FDM arrangement but the digital filter 6(m), m = 1, 2,.~.. N can be ~
considerably simplified. ~ -Figure 30 shows a TDM-FDM arrangement for N = 8 wherein the matrix A(8) on which the trans- ; , formation device 3 is based satisfies the above-mentioned property and thus enables, for example, the fast implementation shown in Figure 27 with ; two-point transformers. In this Figure 30 elements which correspond with Figure 21 and Figure 27 have been given the same reference numerals. This TDM~
FDM arrangement is provided with eight input chan-nels 1(k) with k - 1, 2, 3, ..... N(=8). The input signal9 {xq(n)3 with q = 1, 2, ....... . . . N(=8) are applied to these input channels in an appa- ~-;
rently arbitrary sequence. The sequence of these input signals is chosen such that an FDM-signal is obtained having exactly the same shape as the FDM-signal which is produced by the arrangement of Fi-gure 21 (c~. Figure 2~b). As in Figure 21, the in-put channels to which the input signal.s with even ;' ~
\
3 4 10 6-~7 , ~ .
number (q = even) are .applied are each provided with a sideband interchanging.modulator 2(p) with p =
1,2~3~4. These input channels are connected to the inputs of a fast transformation device 3 which is -eonstructed in the manner shown:in Figure 27 and :~
which produces the complex output signals ~sm(n)~
with m = 1, 2, .... N(=8). These output signals are .
applied to N signal channels. Each signal channel eomprising means for increasing the sampling rate . ~.
: 10 by a desired factor. Contrary to the signal chan- -: :
nels of Figure 21 the signal channeIs of Figure 30 ~ -. .
: are partly used in common, so that each signal eharmel may be eonsidered as having been built up -~
.:
from a plurality of sub-charmels. In the Figure the ~fir~st channels are indicated by 21(.), the seeond sub-eharmels by 22(.) and the third sub-channels by 23(.). The signal eharmel w~eh in Figure 21 is in- ;:~
dieated by 4(1) is now formed in the arrangement of Figure 30 by the series-connected sub-ehannels ;.~.
21(1), 22(1) and 23(1). Likewise, for example the ;.
signal ehannel which in ~igure 21 is indicated by 4(2) is now constituted by the series-eonneeted sub~ehannels 21(2), 22(1) and 23(1).
. ~ : : .
The sub-ehannels 21(.) of Figure 30 each eomprise a series arrangement of an SRI-element 24(.) and a digital filter 25~.). The sub-channels 22(.) eaeh comprise a series arrangement of an adder - 69 - :
~..
10.6.77 :~
1~14~3~ :
~,.
26(.), an SRI-element 27(.) and a digital filter 28(.).
The sub-channels 23(.) each comprise a series arrange-ment of an adder 29(.~, an SRI-element 30(.) and a digital filter 31(.). In addition the sub-channels ;
23(,) are connected to inputs of an adder 32 which produces the desired digital FDM-signal ~y(n)~
Because the transformation device consists of two-point transformers the increase factor of all SRI- :
elements is equal to 2. ~.
In Figure 30 the transfer functions of the various digital filters are indicated by H(1)(~
N(12) (~J), H21)(~), H212)( W) and so on. In general ~`
such a transfer function will be indicated by Hiz)(~
-: -~ hereinafter wherein i = 1,2,3,..7; j = 1,2; and ~ :
- .
wherein z represents the number of the sub-chan-nel z = 1,2,3.
~ecause the digltal signals which at least are applied to the digital filters 25(.) and 28(.) occur with a considerably lower sampling rate then :
the digital signals which are applied to the digi.-.-tal filters 6(.) of the arrangement of Figure 21 ;
the transfer functions of the first-mentioned fil- ; .~
ters can be realized in a considerably simpler man- . .-ner.
In order to obtain the desired ~DM-signal at the output of the adder 32 the transfer functions of the signal channels and the matrix A( ), whose 10,6.77 $$~i34 ~ :
..;~
fast implementation is sho~n in Figure 30 must satis- ~
fy the FDM-condition (41). : -The transfer function of a signal cha.nnel ::~
is now given by the product of the transfer function ~ ~
of the various digital filters which are found in ~:
the successive sub-channels which together constitute the relevant signal channel- So, the transfer func-tion of the first signal channel is, for example, equal to:
Hl(W ) = H1~ J) H(1) (~).H(3)( ~) that of the seoond signal channel:
2(~J) = H(1)(~J ) 'H(1)(~J) (1) that of the third signal channel: .
H (~) = H(1)(~J) . H(2)(~) .H73 ( W).
.15 The fundamental interval of these transfer fu.netions H1(~V), H2(~J), H3(W )... H8(W ) is equal to 8. 2T . If for these transfer funetions, in ae- -eordance with expression (46), the matrix H(~o) is defined again then the FDM-eondition (48) must again be satisfied (M=8).
In the embodiment shown in Figure 30 this FDM-eondition is satisfied by having the transfer-funetions Hiz) (~ ) and Hiz)(~) as well as the ~14534 PHN 8731 '` ,' matrix Aiz)satisfy the FDM-condition. To be able to indi- -cate this more concretely we define the filter sub-matrices~
~ Hiz)(~z) Hil)(~ +2Z ~
Hiz(~z) ( 2 ~ (72) ~.
1z ( z) H(2)(~ +2Z 12~
H ( l ) ( 2 Z+l ~ _ ~ ) H~ ( 2 Z+1- 2z )~ -~Jz ] "~'' : ''''` ~;
Hiz(2Z+l ~ -~z)= ( ) \H(2)(2Z+1 -~ ) Hi2)[(2Z+1-2Z l)~r-~z]/
- .... . .
wherein i = 1, 2, 3, ... , 7; z = 1, 2, 3. .. ~
10~ 2Z-l ~ ,~ ",~,, z T `` ~
` ~ . :
T fsz Herein fsz represents the sampling rate associated with the input signal of the digital filter in the zth sub-channel and :~ .
which has the transfer function Hiz)(~J). Again T represents ~.
15 the sampling rate associated with the signals {sm(n)}. So it .
holds for the filters 25(.) that z = 1 and, consequently, :`-fsz = fSl = T. It likewise holds for the filters 28(.) that z = 2 and for the filters 31 (.) that z=3. ~;
By means of the filter sub-matrices (72) :
~ ! i - 72 - ~
' ~ , ~4~ 10.6.77 ~ ' ; and (73) the FDM condition (48) now changes into:
~iz Hiz(~Jz) + (A(z~ iz(2 ~T ~ ~Jz) =~2I2 (74) The sub-channels 21(.), 22(.) and 23(.) are now all implemented in the manner as indicated in Figure 28. The index m which is used in Figure 28 for the denomination of the transfer functions must now be replaced by a combination of indices i, z and ~. This means that the digital filter HiJ)(~) is now built-up by means of the digital filters HiJ)( W) and Hi3)(~) which are given by (603. For the additional conditions specified in chapter E(2.5.2) the general implementation indi-cated in Fi~ure 28 of the sub-channel changes into ~ that of Figure 28a or 28b or 28c.
1~ It should be noted that in the embodiment ;
shown in Figure 30 the fast implementation of the transformation device was based on two-point trans- -formers each having associated therewith a 2x2-matrix Ai2). If~ however, this fast implementation was based on four-point transformers each having associated therewith a 4x4-matrix Ai4) then the above holds similarly.
E(? 8~ Transformer matrices and transfer functions.
From the above it will be clear that a great number of transformer matrices is suitable in ~ :
principle to sérve as the base for the transformation device. However, not all transformer matrices which . . .
10.6.77 ~L4~3~ :
:,~
,.
~,.
are suitable in principle will result in a realizable TDM-FDM arrangement because they will lead to an ex-cessive number of multiplioations in the transforma-tion device or ts very complicated digital filters.
In this chapter a number of matrices will :~
be described by way of example, which may result in simple digital filters as well as in a simple trans-formation device and which will allow a fast imple-mentation of the transformation device.
A class of matrices which satisfies the `~
above requirements is formed by the Hadamard matrices `
; ~ where ~ = 1,2,3,.... which are defined as fol- -lows:
.: .
:: ~ .:.
, ~ . .... .. . . .. _ ... _.; .. _ __ . .. ..... ........ .... . .. . ..
J~ -1 ~V ~ . ~ V =1' . .~ _ _ :: .
:, ~:
wherein ~ = 2~ 3~ ... and where ', ,...: :-~1 1 , ,.
01 = ~ (76) -:' A Hadamard matrix is a real matrix which satis-fies (68).
Because a Hadamard matrix allows a fast implementation only the simplificat~.ons will be indicated here which are possible in the TDM-FDM
, P}IN 873i 10,6.77 1~4534 ; - ~
', . :~` ' arrangements shown in Figure 30. : -In the arrangement of Figure 30 the number :~ -~: , - . .
:: of input signals N is equal to 8 = 23. This méans . . ; ~
that the transformation device 3 must be based on - :
the 8x8 Hadamard matrix 03. This results in that:~
Aiz) = 01 for all i and all z (77) :~
. Now as an additional condi.tionit can be chosen: ;
; . Hi ( W ) = ~iz(2 T ~ W
' 80 that Hizq( z) ~ :.
;~ : . . , ~ ~ . From (67) it now follows that: :
~ ~ HiZP(~JZ) = L01T] = 2 (1 1) for all i and all z ~;~ N (zzt1 ~r ~ ) _ ~ ~ T~ -1} (78) 2 (1 -1) for all i and all z : ':
' From (76) and from Figure 23 it follows that the two-point transformers which together con- ~:
, stitute the transformation device of Figure 30 are I ;-I ,.
very simple. Because the multiplication factors ~ :
indicated in Figure 23 are now given by: ¦ ;
~,, .
: ~ 75 10.6.77 534 .
,..~
: ~ 11 = ~12 = ~21 =
22 = -1 12 = ~21 = ~ 22 =
the multipliers 9(1), 10(1) and 9(3) o~ Figure 23 -~
oan be replaced by through-colLnections~
~: : Before the influence of the Hadamard matrix ..
:
on the transfer functions of the digital ~ilte~ is --`, ~: described it is be noted that the TDM-FDM arrange-; ment shown in Figure 30~comprises seven circuit 10~ configurations of the type depicted in Figure 31. .
This circuit configuration comprises two channels ~ .
I and II. To each of these channels a digital sig- ~
nal having associated therewith a sampling period T is applied and this configuration produces a : .-: : 1S digital output signal having associated there- : :
!- - :
with a sampling period Tz, For the sub-channels ~ .' . ~ 21(.) in Figure 30 z = 1, for the sub-channels . 22~.) z = 2 and for the sub-channels 23(.) z = 3. ~--By choosing .the Hadamard matrix as the ~.
transformer matrix it is now obtained that: :
~: -:
~ Hiz(~ ) = H (~V
.
so tha.t the circuit configuration shown in Figure ~ -;
31 changes into the circuit configuration depicted in Figure 32 (see also Flgure 28c). .
"
. - 76 .~ . .
:~:
\
5 ~ ~ 10-6-77 1. ' .
, From (78) it fo~ws that:
~( )( W ) = H(lP ~JZ~Z T ) = 1 H(l) (2Z~ W ) = H(1) [(ZZ~l ZZ-~
1.
so that the digital filters having the transfer func-tions Hiz)(w ) are each equivalent to a through- 1 connection. For completeness this situation is de-picted in Fig. 33. Consequently the TDM-FDM arrange-ment comprises only N-1=7 digital filters.
;10 From (78) it furthermore follows that ~or ! ` -~
the digital filters having the transfer functions ~ ~
Hizp(~V) it holds that: ;
Hizp ( W z ) = H ( 2 ) ( 2 ~+1 '17' izp Z T ) izp ~ ) T -~ ~z]
i =-1.
This means that these digital filters are formed by all-pass filter which introduce a phase shift of 0, - ~ or ~ ~ . Figure 34 shows, for completeness, the argument of H(2) (~J) as a function of ~
A further simplification of the circuit configuration of Figure 33 is depicted in Figure 35. This configuration differs from tllat of Figure ,. ~ 10.6.77 $;~ `
..
33 in that the channel II is now constituted by a se-ries arrangement of a digital filter having a trans-fer function Gz,- the SRI-element having an increase factor 2 and a delay device having a delay time zZ .
The amplitude-frequency characteristic of both the ~ r ; digital f'ilter and the delay device is equal to -unity. Figure 36 shows two fundamental intervals ;
of the phase-frequency characteristic of the digi~
tal filter. For completeness Figure 37 shows the phase-frequency characteristic of the delay device s~
over one fundamental interval. The phase-frequency chzracteristic shown in Figure 34 is now obtained by adding the characteristic shown in the Figures ~ ;
36 and 37 together. ~ -In a practical embodiment the circuit con-figuration Or Figure 35 can be reduced to the confi-guration depicted in Figure 38 and which is equi-valent to the configuration depicted in Figure 35 and which is also provided with two channels I and - ~
. ~ ,. .
II. The functions of the SRI-elementsJ the delay device and the adder shown in Figure 35, however~
are now concentrated in a switching device 33. This switching device, which is only symbolically shown, ;
is controlled by switching pulses which are produc- ;
ed with a period z. At the occurrence of the first of two successive switching pulses the channel I ;`
is connected to the output 34 of this switching : .
PlIN 8 7 31 10.6.77 ~$4LS34 - ,,, device and when the second switching pulse occurs the ~.
channel II is connected to this output 34. At the output 34 of the circuit configuration a digital signal is produced having associated therewith a .' sampling period z and;which is'formed by a suc- ' ' - cession of signal components which alternatingly ~' originate from the channel T and the channel II :' (interleaving). ' ;-For completeness Figure 39 shows the-~com- ~ ' plete implementation of the TDM-FDM arrangement ~.
wherein the Hadamard matrix Ai )= 01~ defined in'' ~: (76~, is the basis of each of the transformers.
-This'TDM-FDM arrangement follows., together with ' the preceding description, from the Figures 30, 23, .' -~
15 ' ' 27'and 38. The'digital filters, having the trans-fer functions G1, G2, G3 are all all-pass fiiters having the phase-frequency characteristicsshown in' ~igure 36. ' :~
A further real matrix which results in '~ ' 20 both simple digital filters and a simple transforma- ` .~ :
tion device and which allows a fast implementation of this transformation device is obtained if ~.
. ~ 1 0\
Aiz2) = ~ ¦ for all i and all z (79) .
~_1 1/ :
If it is again assumed that: Hi ~ ) = O it then follows together with (74) that:
~ 79 -,~.
! PHN 8731 10.6.77 '-~
I
- /~ 1 : ~
-iz(~Jz) = Hizp( Z) 1 for all i and all z :~
0 1 ' (80) H (2Z+l T~ ~ ~) ) = HiZp(2æ+l~T ~~Z) = ( ~ ) '~
; ~ 1 ~
Also with this choioe of the transformer matrlx ~,' '-;
Aiz the circuit configuration of Figure 33 applies, so that the TDM-FDM arrangement of Figure 30 can !
again,be realized with only N-1 digital filters.
. ,~
. The digital filter shown in Figure 33 '.- :
, with transfer function Hi2z)(~J) can now again be ,~:, realized by a cascade arrangement of a digital fil- ~
ter with transfer function Dz(h~) and a delay de~ice ".,'.~.
having a delay time Tz (see Figure 40). The amplitude- ,':'.
frequency characteristic of Dz(W ) is shown in Figure ~
41 and its phase-frequency characteristic is shown `~ ,., in ~igure 42.
' ~ When the transformer matrix is chosen ac-cording to (79) and when a phase distortion may be allowed, the circuit configuration of Figure 33 can also be implemented in a manner different from the implementation which is depicted in Figure 40. Tak- ~ -ing into account the remarks indicated in chapter , E(2.3), the FDM-conditlon (74) CQn now be written in the form: .
.
, - 80 - ~ `
1~45~ lo. 6.77 ~.
", ) Hiz (W ) = 2I2 diag ~ ~iz(~ z)] ~ ;
~' so that:
Hizp(h/ z) = ( )(,j~iz; 1 ( ~z) 0 O 1 0 yiz;2(~z) 1 .
iz;1(~z) ~iz;2( ~ z)~
. :, . ~iz;2(~ z) ::
Because Hizp(~J ) = O a phase-frequency function ~ ;
may be allotted to Hi2)(~Jz) which is equal to yi 1(~Z) If it is now assumed:
; izp ¦ z(~ )¦ ~iz,2( W ) whereby the f~mction¦D~ is depicted in Figure . I
41~ then we are stlll free to choose ~ iz;1(~z) equal to ~ iz;2(~Jz)' This means that in ~igure 33 a digital ~ ter having the amplitude-frequency :.
function shown in Figure 4t and a phase-frequency function which is given by ~i ~2( W) must be in-cluded in channel II and that a digital all-pass filter having the saMe phase-frequency function ~iz;2(W ) must be included in the ch~nel I.
In chapter E(2.4) it was noted that I :
a fast transformation device can also be implemented i -~
' 81 - - ! :
, . . I
.~ . . .. . . ...
1$~3g 10.6.77 ~ -:~ -: - -, by means of, for example, four-poin.t- transformers. j : :
Such an implementation can,. for example, be advan- ~ ::
tageous if N=2 wherein ~ represents a positive integer, With such an implementation the increase ~ -.
factor of SRI-elements in the various sub-channels . --of the device of Figure 30 is equal to four and ~ ~
the transfer matrix i9 defined for four sub-channels . ;:
of the same number. For completeness a real trans- i~
former matrix will now be given which might be the -::
~ ::
; lO basis for the four-point transformers. Such a matrix, ~:; for example, is the real matrix .~ :;
/ 1 1 o : ~ A(4)= 0 1 -1 0 for all i and all iz O 0 1 -1 z : `.
150 0 0 1 .
If now a8ain the additional condition is given by:
then it follows from (69) that:
Hi ( W ) = (Ai4)T) ? O so that:
11 0 , O - O \
--izp(~'~
~ - 82 t `': " ' ;~
PHN (~731 ~ 4 1o. 6.77 In chapter E(2.7) it is indicated that, the TDM-FDM-arrangement depicted in Figure 21, can be , replaced by the TDM-FDM -arrangement depicted in Figure 30, when the transformer matrix allows a fast implementation. In case W 1= and the transformation - device is based upon a real Hadamard matrix the ar-rangement of Figure 30 can be still further modified to the arrangement shown in Figure 39.
If ~J1 ~ the signals {rk(n)} applied to the transf~mation device in the arrangement of Figure 30 are complex. This means that the increase factor of the SRI-elements in the various sub-chan-nels must at least be equal to three. From Figure ; 30 it will be clear that this means that if with bhe arrangenentof Figure 30 eight signals {xk(n)}
which are all unequal to zero must be converted in-to a FDM-signal a sampling rate must be associated with the output signal {y(n)} which is at least equal to 27/T. If this is compared with the sam- -plin~ rate whic~ is associated with thesignal f y(n)} which is produced by the arrangement of Figure 21 and which is at least equal to 9/T
then it appears that the implementation shown in Figure 30 is not attractive if hJ1 ~ -If for special circumstances, for example, if t~le signals` frk(n)3 represent complex signals, in the arrangement shown in Figure 30, a complex -;
:
- ~ r PH.N 8731 10.6.77 ~-~$~
transformer-matrix is preferred over a real transfor-mer-matrix then it would, for example, be possible . to take therefore the ma~rix~
/l~j l-j\ .
Al2) = 1 ~ ) for all i and all z ;~
l-j -l~J , To indicate the influence of this choice of the trans- ~`
former-matrix on the digital filters we shall again restrict ourselves to the case ~J1 = 0. I~ the ad-ditional condltion:
Hiz(2 T ~ ~Jz) =
is now chosen then the sub-channels must be construct-ed in the manner shown in Figure 28a. From (65) it now follow~ that: ~-/1 l\
HizP(4J z) = 2 ( ) for all i and all z _,~ , . , --i Zp ( 2 1 ~ .
As regards the transformation device the above means that~
_ 84 10.6.77 ;
3~ ;
. ~ .
~11 = 1- ~ 12 = 1 12 = -1 21 = 1 CX 22 = -1 `
~ 2~ 22 = 1 5 . So it applies for the digital filters HiJp(~ ) that the filters having the transfer function Hizp(w ) ! ;~
just form through-connections and that the digital filter having the transfer function Hi2)(~J) forms an all-pass filter which introduces a ~ 2~ phase .
10 ~ shift in the interval 2 l~r ~ ~ C 2Z 1r and which introduces a + 2 phase shift in the interval T ~ ~2 i ~1T . For completeness ~igure 43 .
shows the argument of Hi,2,)(~) as a functioll of~J .
: .As has already been remarked, if N = 4 , j:~
15 . wherein ~ represents a positive integer, the trans- ;~
formation device can also be based upon four~point transformers .
A possible 4x4-matrix is the complex matrix !
1 1 1 1\
Aiz ~ ~ with J ~ (81) - j - 1 j `''' for all i and all z.
With the additional condition: ./
Hiz(2 ~ _kJ ) = o ~:
' - 85 - .
~ 4 10-6-77 ~, ;
, ~ .
: , .
~ it follows from (65) that:
/1 1 1 1 . ~ . , (~U ) ~ with j = ~ (82) for all i and all z.
In full agreement with the above the transfer functions i1 ~ ;, in the various intervals follow from these transfer- j matrices.
For completeness Figure 44 shows an elaborat-~~ ed embodiment of the arrangement of Figure 21 wherein the real signals ~xk(n)} are converted into complex signals by means of the complex modulators. In the arrangement: of Figure 44 N = 4, M = 5 and the transformation device is based upon the matrix Aiz) given in (81). For the transfer functions of the digital filtqrs it is again assumed that Hm (5~ J0) = 0, so that the signal channels must be implemented in the manner shown in Figure ~ I
28a. The transfer functions Hmp(~) of the digital , ~ -filters incorporated in these signal channels are ~ -given by (82). The amplitude-frequency flmction is the same for all digital filters Hmp( ~ ) with m = 1,2, 3,4 and is shown in Figure 45.,'rhe phase-frequency functions of the various digital filters H p are depicted in Figure 46, '~, .
_ ~6 , 1~3L4S34 1 o . 6 . 77 .'; :, ` As can be easily checked, taking into ac-count the expressions (16), the TDM-FDM arrangement shown in Figure 44 produces the FDM-signal shown in ;
Figure ZOa.
It should be noted that in the arrangement of Figure 44 only the real part of the complex sig- -nals ~um(n)} is determined. For, the imaginary - ~
part of this ~ignal has no oontribution to the de- ~-~ "
sired FDM-signal. 7 E(2.9) General remark~about the TDM-FDM arran~ement Each of the Figures 21, Z8, 28a, 28b, 28c, ~ ;
.
30,31,32,33,40 and 44 show series arrangements of an SR} element and a digital filter. In a practical em-bodiment of such a series arrangement the function ~ ~ Or the SRI-eIement and that of the digital filter are interwoven so that a practical embodiment o~
uoh a series arrangement is constituted by an in-terpolating digital filter which is also called a :, ~ ~ sampling rate increasing digital filter. For the ~ -; 20 implementation of such a digital filter we refer, rOr example, to references 15, 16 and 17. ~
2, The elements 8 shown in the Figures 21 ``
and 30 are used for mathematical purposes only. As ~
. .
appears from the Figures 39 and 44 such an element -~
is not used in a practical embodiment of the TDM-FDM convertor because the real and the imaginary ~
part of a complex signal are available separately. ~ -~ ~' PH~ 8731 10,6.77 , To determine the signal Re L v(n)~ it is sufficient to apply the signals Re ~ um(n)] to the adder 7 and 32 respectively.
3. In the preceding it was assumed that, if W 1 = the channel signals in the frequency spec~
trum of the FDM-signal are located as shown in Fi-gure 20b. This means that for N = 4 the FDM-signal of the four baseband signals ~ x1(n)~ ' {X2(n)}
~x3(n)} and ~x4(n)~ is located in the frequency band of 0< ~ 8T/ . However, if it is desired to have the FDM-signal situated, for example, the fre-quency band ~C 9 ~ then, of course, ~J1 may be ~
taken equal to T . It is, however, simpler to choose ~ `
N equal to 5 and to compose a FDM-signal starting ~ -from the five baseband signals ~ xO(n)} ~ {x1(n)~
2(n)3 ~ {X3(n)~ and fx4(n)3 ~ whereby fxO(n)~ is equal to zero for all n.
4. If ~V 1 ~ ~ as in the TDM-FDM arrangement of Flgure 44 the transfer functions Hm(~J) must be equal to zero in those frequency intervals which are not covered by the FDM-¢ondition. --F(1~ The FDM-TDM arranFement.
Chapter E extensively describes arrange-ments for converting a plurality of discrete base-band signals into a discrete baseband single-side-band frequency-multiplex sign~l. This chapter briefly deals with arrangements formed by the trans-' . - 8~ -PH~ 8731 ~4~3~ 10.6.77 .
posed configurations of the arrangements described .
in chapter E and which thus convert a discrete base-division band single-sideband frequenc ~ multiplex signal into the original, spatially distributed, discrete base-band signals. The transposed configuration of a given arrangement is obtained by - reversing the direction of all signals;
- replacing the adders by branch nodes;
- replacing the branch nodes by adders;
- replacing the SRI-elements by SRR-elements;
- replacing the SRR-elements by SRI-elements. .
; The transposed configuration of the TDM-; ~: , , . ' .`' - FDM arrangement shown in ~igure Z1 is depicted in Figure 47. This FDM-TDM arrangement comprises a plurality of N signals ohannels 4'(m) with m = 1,2,3,N.
Applied to each of these signal channel~ is the FDM-signal fy~n)~ having the frequency spectrum y( W) which, for example, again has the form which for N- 4 is depicted in Figure 20a is applied to each signal ~:
channel. Each signal channel 4'(m) comprises a di- ~
gital filter 6~(m) having the transfer-function E (h~) ~ :
and an impulse response em(n), which, in general is complex em(n) - Re L em(n)~ + j Im [em(n)~
This di~ital filter 6'(m) produces a digital out-put signal t'~n(n) which is given by:
t'm(n) = Re ~ t m(n)] + J Im rt m( )~
PHN 8 7 31 ~:
34 1o. 6.77 If` ~y(n)~ represents a real signal it follows that (see 52)) ; [ m(n)~ = y(n) x Re ~e (n)~
: L [ ~ ::
The frequency spectrum T'~(J) of {t' (n)~ is given by:
Tm( W ) = y(~ Em~ ) . This signal ~t' (n)~ is applied to an SRR-element 5'(m). From (11) it follows that the components~
~10 5~m(n) of theoutput signal fs~ (n)3 of the SRR-element are equal to~
(n) = tm(Mn) From (12) it follow~ that the frequency spectrum S~ ) is given by~
Sm(~) c M ~ T'm ~+(q~1) wherein T represents the sampling period associated with the signal { s'm(n)~ :
The signals { s'm(n)} thus obtained are applied to the transformation device 3' which com- -prises N output channels 1(k) and whose operation can again be fully described by means of a matrix B = r bkm~ comprising the elements bkm. This trans-' _ 90 -PHN 8731 : -10.6.77 :
, formation device produces N digital signals -:
~k(n)~ , k = 1, 2, 3, ..... N, which are given l, .
by:
N . ~:
rk ( n ) = ~ bkm Sm ( ) ~ ~:
The associated frequency spectrum R'k(~) is given ~ :
by~
Rk(~J) = ~ bkm Sm( 4 ) , The complex signal {r'k(n)} is applied to a com- '- :
plex demodulator 1'(1,k) which produces the complex signal ~wk(n)} which is given by wk(n) = r'k(n)e j lnT.
Of this signal the arrangement 8(k) now produces the -real part of the signals fwk(n)3 . Th~ the out-put signal vk(n) of this arrangement 8(k) is given by v (n) = Re rwk(n)~ = ~ rwk( ) k Herein wk ~(n) again represents the complex conjugate of wk(n). For the frequency spectrum Vk(~J) of fvk(n)~ lt again holds (see (30)) ~ Vk(~V) = 2 fRk(~J + W 1) + ~ [ T ~~ +~J1]~
For the output channels l(k) with odd nwnber k it again holds that xk(n) = vk(n) and each the . .
output channels with even num'~er k again comprises ~ :
.
PHN 8731 - :~
1~4S3~ 1 o .~ . 77 '~
'`
a sideband interchanging modulator 2'(.) whose ,.
output signals constitute the desired signals k(n)} for k is even. ~ `
After some manipulation in the manner in-dicated in chapter E(2.3~ it is possible to derive the.TDM-conditions from the~above expressions. The .
. TDM-condition for ~J1 - O as follows: ~
. _ _ ,, .
-, . ..
m=I { km m ~Jo + (i-l) ~ ] 4 .'~
kmEm~ 0 ~ 1) T ¦} = S (83) :: ~ 10 The TDM condltion for ~Jl ~ is as follows: ;- "~:~
Em ~M T ~ WO + ~J1 + (i-1) T~ =
~ ~ ~84).
: ~ 1 N
~1 km m ~VO + ~)1 + (i-1) T-3 = ~ ki ::
In (83) and (84) ~ ki again represents the Kronecker symbol which is defined in (43). Furthermore it ap- -~15 plies that: :`
o~ w ,~
.-.
.i = 1, 2, 3, ... N.
Just as the FDM-conditions (41) and (42) the TDM con-' 10.6.77 ;
1~4534 :-ditions (83) and (84) can be written in the form of ¦
a matrix, the matrices defined in (46) and (47) be-ing applicable so that (83) changes into:
B(N) E(~J )+B(N) E (M.2 ~ _ ~U0) = 2NIN.diag ~ i(4Jo)~ (85) ~ '~
and (84) changes into:
E (M T ~ ~o) =
(86) ;
; B( ~ E(~lo) = 2NI~-diaF [ ~i`( ~ )~
Figure 47 shows a possible implementation ~ of a FDM-TDM arrangement wherein the relationship ;~ between the transfer functions Em( W ) of the digi-tal filters 6~(m), with m =1, 2, 3, ... N and the transformation matrix on whioh the transformati~n devioe 3' is based i9 given by the TDM-condition (84) or (86). As for the TDM-FDM arrangement a oom-plete freedom in the choice of the various matrices is possible so that what has been described for the ~-TDM-FDM arrangement fully applies for this -FDM-TDM ~ -arrangement.
Comparing the TDM-conditions with the FDM-oonditions shows that these conditions are not iden-tioal. In the oase the FDM-TDM arrangement is ob-tained by transposing the ~DM-FDM ~rrangement this .: ' , PI~N ~731 - ~
10.6.77 .-;: . .'"
is, however, only seemingly so. By transposing the ... .... ......... .... digital filters 6(m) Or Figure 21 these filters change into the digital filters 6'(m) of Figure 47.
It is true that this causes the implementation of . -the digital filter 6'(m) to be different from that of the filter ~6(m), but the transfer function of the filter 6(m) is not affected hereby (see reference 2) so that: . .
Em(~J) = Hm~J) . (87) It can be checked ln a simple manner that by trans~
: : posing the transformation de~ice 3 which i8 based on a matrix A( ) = ~ a k] the transformation de-. vice 3' is obtained which is based on a matrix A( ) = . .-[akm] so that 9(N) = A(N)T (88) ..
; Substitution of (87) and (88) in (86) again fur- .:
nishes the FDM-condition (49) and substitution of (87) and (88) in (85) furnishes the FDM-condition (48). From the preceding it consequently follows that a FDM-TDM arrangemant can be obtained by trans- `~
posing a given TDM-FDM arrangement or vice versa.
F(2~ General remarks about the FDM-TDM arrangement :: 1. Figure 47 and the arrangements which are obtained by transposing the TDM-FDM arrangements ::
shown in the Figures 30 and 39 each comprise series ~
_ 94 - , ,. .. l- .- - , ... .. . .
10.6.77 ' arrangements of a SRR-element and a digital filter.
In a practical implementation of such a series ar-rangement the function of the SRR-element a~d the function of the digital filter are interwoven. Such g a series arrangement is consequently implemented by means of an extrapolating digital filter, also called sampling rate reducing digital filter. For the implementation of such a digital filter we re-fer to the references 15 and 18.
2. The complex demodulator shown in Figure 47 can, as the further elements of this arrange-ment, be obtained by reversing, in Figure 17, the ~
- signal direction and by replacing the distribution ~ -point by a subtractor. With such ~ implementation of the complex demodulator the output signal here-of is given by Re [wk(n)3 so that the arrangement 8(k) can be dispensed with in a practical embodiment of the FDM-TDM arrangement.
3. The TDM-FDM arrangement and the FDM-TDM
arrangement need not form a system in which, for example, the TDM-FDM arrangement functions as a transmitter and the FDM-TDM arrangement as a re-ceiver or vice versa. Each of these arrangements can be used in existing PCM telecommunication sys-tems independent of the application of the other arrangement.
~ 95 , ~ .. ,.. . , .. . .. ,. . . .. ,,, !.'.. '.-. . .
Claims (5)
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:
1. An arrangement for converting a discrete baseband single-sideband frequency-division-multiplex signal {y(n)}, n = ... -2, -1, 0, +1, +2,....) having associated therewith a sampling rate ? which is at least equal to ?, said signal {y(n)} being formed by N channel signals and having a frequ-ency spectrum Y(.omega.), into N discrete baseband signals {xk(n)}, (k = 1, 2,...N) having associated therewith a sampl-ing rate ?, which baseband signals are representative of said channel signals and which each have a frequency spectrum Xk(.omega.), wherein Xk(.omega.o) = Y [.omega.o+ (k-l) ?] ?? k(.omega.o), characterized in that I. the arrangement comprises:
- means for receiving said discrete frequency-division-multiplex signal {y(n)} ;
- a plurality of signal channels to each of which the said discrete frequency-division-multiplex signal {y(n)} is applied and which are each provided with discrete filtering means and sampling rate reducing means for generating dis-crete signals {sm(n)}, the transfer function of the signal channel being determined by said filtering means and being equal to Em(.omega.);
- a transformation device to which said discrete signals {sm(n)} are applied and which is arranged for processing these signals for generating a plurality of discrete signals {rk(n)} said transformation device having associated there-with a transformer matrix B comprising matrix elements bkm of a constant value, said transform matrix being unequal to an inverse discrete fourier transform matrix and whereby the relationship between the components sm(n) and the components rk(n) is given by - an output circuit to which the signals {rk(n)} are applied and which is provided with means for selectively modulating the signals {rk(n)} for generating said discrete baseband signals {xk(n)};
II. that for each signal channel the relationship between its transfer function Em(.omega.) and the matrix elements bkm is given by the TDM-condition wherein:
m represents the number of the relevant signal channel;
.omega.o represents a frequency in the range O?.omega.o<?;
represents the complex conjugate of bkm ;
E?(.omega.) represents the complex conjugate of Em(.omega.) ;
i = 1,2,3,...N;
?ki = O for k ? i ?ki = l for k = i;
?i(.omega.o) represents an arbitrary function of .omega.o.
- means for receiving said discrete frequency-division-multiplex signal {y(n)} ;
- a plurality of signal channels to each of which the said discrete frequency-division-multiplex signal {y(n)} is applied and which are each provided with discrete filtering means and sampling rate reducing means for generating dis-crete signals {sm(n)}, the transfer function of the signal channel being determined by said filtering means and being equal to Em(.omega.);
- a transformation device to which said discrete signals {sm(n)} are applied and which is arranged for processing these signals for generating a plurality of discrete signals {rk(n)} said transformation device having associated there-with a transformer matrix B comprising matrix elements bkm of a constant value, said transform matrix being unequal to an inverse discrete fourier transform matrix and whereby the relationship between the components sm(n) and the components rk(n) is given by - an output circuit to which the signals {rk(n)} are applied and which is provided with means for selectively modulating the signals {rk(n)} for generating said discrete baseband signals {xk(n)};
II. that for each signal channel the relationship between its transfer function Em(.omega.) and the matrix elements bkm is given by the TDM-condition wherein:
m represents the number of the relevant signal channel;
.omega.o represents a frequency in the range O?.omega.o<?;
represents the complex conjugate of bkm ;
E?(.omega.) represents the complex conjugate of Em(.omega.) ;
i = 1,2,3,...N;
?ki = O for k ? i ?ki = l for k = i;
?i(.omega.o) represents an arbitrary function of .omega.o.
2. An arrangement for converting a discrete baseband single-sideband frequency division-multiplex signal {y(n)}, (n = .., -2, -1, 0 +1, +2,...) having associated therewith a sampling rate ? which is at least equal to ? wherein M is an integer, said multiplex signal being formed by N
channel signals wherein N is smaller than M and having a frequency spectrum Y(.omega.), into N discrete baseband signals {xk(n)}, (k = 1,2,3,...N) having associated therewith a sampling rate ?, which baseband signals {xk(n)} are repre-sentative of said channel signals and which each have a frequency spectrum Xk(.omega.) wherein Xk (.omega.o) = Y [.omega.1 + .omega.o + (k-l) ?]??k(.omega.o) and wherein .omega.1 ? ? ? ?
where ? = 0, ?1, ?2,... , characterized in that I. the arrangement comprises:
- means for receiving said frequency-division-multiplex signal {y(n)};
- a plurality of signal channels to each of which the dis-crete frequency-division-multiplex signal {y(n)} is applied and which are each provided with discrete filtering means and sampling rate reducing means for generating discrete signals {sm(n)} , the transfer function of the signal channel being determined by said filtering means and being equal to Em(.omega.);
- a transformation device to which said discrete signals {sm(n)} are applied and which is arranged for processing these signals for generating a plurality of discrete signals {rk(n)}, said transformation device having associated therewith a transform matrix B comprising matrix elements bkm of a constant value, said transform matrix being unequal to an inverse discrete fourier transform matrix, and whereby the relationship between the components sm(n) and the com-ponents rk(n) is given by - an output circuit to which the signals {rk(n)} are applied and which is provided with a cascade circuit of selective modulation means and complex modulation means with which a complex carrier signal having the frequency ? is associated for generating said discrete baseband signals {xk(n)};
II. that for each signal channel the relationship between the transfer function Em(.omega.) and the matrix elements bkm is given by the TDM-condition E? [? - {.omega.1 + .omega.o + (i-1) ?}] = 0 wherein:
m represents the number of the relevant signal channel;
.omega.o represents a frequency in the range O ?.omega.o < ?;
E?(.omega.) represents the complex conjugate of Em(.omega.);
i = 1,2,3,...N;
?ki = 0 for k ? i ?ki = 1 for k = i;
?i(.omega.o) represents an arbitrary function of .omega.o.
channel signals wherein N is smaller than M and having a frequency spectrum Y(.omega.), into N discrete baseband signals {xk(n)}, (k = 1,2,3,...N) having associated therewith a sampling rate ?, which baseband signals {xk(n)} are repre-sentative of said channel signals and which each have a frequency spectrum Xk(.omega.) wherein Xk (.omega.o) = Y [.omega.1 + .omega.o + (k-l) ?]??k(.omega.o) and wherein .omega.1 ? ? ? ?
where ? = 0, ?1, ?2,... , characterized in that I. the arrangement comprises:
- means for receiving said frequency-division-multiplex signal {y(n)};
- a plurality of signal channels to each of which the dis-crete frequency-division-multiplex signal {y(n)} is applied and which are each provided with discrete filtering means and sampling rate reducing means for generating discrete signals {sm(n)} , the transfer function of the signal channel being determined by said filtering means and being equal to Em(.omega.);
- a transformation device to which said discrete signals {sm(n)} are applied and which is arranged for processing these signals for generating a plurality of discrete signals {rk(n)}, said transformation device having associated therewith a transform matrix B comprising matrix elements bkm of a constant value, said transform matrix being unequal to an inverse discrete fourier transform matrix, and whereby the relationship between the components sm(n) and the com-ponents rk(n) is given by - an output circuit to which the signals {rk(n)} are applied and which is provided with a cascade circuit of selective modulation means and complex modulation means with which a complex carrier signal having the frequency ? is associated for generating said discrete baseband signals {xk(n)};
II. that for each signal channel the relationship between the transfer function Em(.omega.) and the matrix elements bkm is given by the TDM-condition E? [? - {.omega.1 + .omega.o + (i-1) ?}] = 0 wherein:
m represents the number of the relevant signal channel;
.omega.o represents a frequency in the range O ?.omega.o < ?;
E?(.omega.) represents the complex conjugate of Em(.omega.);
i = 1,2,3,...N;
?ki = 0 for k ? i ?ki = 1 for k = i;
?i(.omega.o) represents an arbitrary function of .omega.o.
3. An arrangement as claimed in Claim 1 or 2, characterized in that said matrix elements bkm are each equal to .alpha.km + j.beta.km, wherein .alpha.km and .beta.km are constants, each having a value associated with the set of values 0, +1, -1, wherein .
4. An arrangement as claimed in Claim 1, character-ized in that each signal channel is formed by a series arrangement of a plurality of sub-channels which are each provided with discrete filtering means and with sampling rate reducing means, and whereby in the zth sub-channel a discrete signal having associated therewith a sampling rate ?, is applied to the discrete filtering means.
5. An arrangement as claimed in Claim 1 or 4, characterized in that said transformation device is formed by a fast transformation device comprising a plurality of transformers each having associated therewith a pxp sub-matrix of a set of pxp matrices (? = 1,2,3,...; z= 1,2,3,...;
.alpha., .beta. = 1,2,3...p) as well as a group of p sub-channels of equal number z and belonging to different signal channels, the discrete filtering means of the sub-channels of number z having the respective transfer functions with j = 1,2,3,...p and whereby the relationship between the elements b?z,.alpha..beta. and such a transfer function is given by the said TDM-condition wherein:
.omega.z represents a frequency in the range 0?.omega.z < pz-1 ?;
b?z,.beta..alpha. represents the complex conjugate of b?z,.beta..alpha.
E??x represents the complex conjugate of E??;
i = 1,2,3,...p;
?.beta.i = 0 for .beta. ? i ?.beta.i = 1 for .beta. = i ?/z;i(.omega.z) represents an arbitrary function of .omega.z.
.alpha., .beta. = 1,2,3...p) as well as a group of p sub-channels of equal number z and belonging to different signal channels, the discrete filtering means of the sub-channels of number z having the respective transfer functions with j = 1,2,3,...p and whereby the relationship between the elements b?z,.alpha..beta. and such a transfer function is given by the said TDM-condition wherein:
.omega.z represents a frequency in the range 0?.omega.z < pz-1 ?;
b?z,.beta..alpha. represents the complex conjugate of b?z,.beta..alpha.
E??x represents the complex conjugate of E??;
i = 1,2,3,...p;
?.beta.i = 0 for .beta. ? i ?.beta.i = 1 for .beta. = i ?/z;i(.omega.z) represents an arbitrary function of .omega.z.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA368,633A CA1114534A (en) | 1977-04-04 | 1981-01-15 | Arrangement for converting discrete signals in to a discrete single-sideband frequency division- multiplex-signal and vice-versa |
Applications Claiming Priority (4)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| NLAANVRAGE7703633,A NL180369C (en) | 1977-04-04 | 1977-04-04 | DEVICE FOR CONVERTING DISCRETE SIGNALS TO A DISCREET SINGLE-BAND FREQUENCY-MULTIPLEX SIGNAL AND REVERSE. |
| NL7703633 | 1977-04-04 | ||
| CA300,017A CA1109973A (en) | 1977-04-04 | 1978-03-30 | Arrangement for converting discrete signals into a discrete single-sideband frequency division-multiplex- signal and vice-versa |
| CA368,633A CA1114534A (en) | 1977-04-04 | 1981-01-15 | Arrangement for converting discrete signals in to a discrete single-sideband frequency division- multiplex-signal and vice-versa |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA1114534A true CA1114534A (en) | 1981-12-15 |
Family
ID=27165589
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA368,633A Expired CA1114534A (en) | 1977-04-04 | 1981-01-15 | Arrangement for converting discrete signals in to a discrete single-sideband frequency division- multiplex-signal and vice-versa |
Country Status (1)
| Country | Link |
|---|---|
| CA (1) | CA1114534A (en) |
-
1981
- 1981-01-15 CA CA368,633A patent/CA1114534A/en not_active Expired
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